(CNN) -- Sen. John McCain told supporters he will "take nothing for granted" after defeating former Rep. J.D. Hayworth in a bitterly fought Republican primary for U.S. Senate in Arizona Tuesday.

"I ... will fight with every ounce of strength and conviction I possess to make the case for my continued service in the Senate, and the policies and principles I will advocate and defend if I'm fortunate to be re-elected," McCain said in his victory speech Tuesday night.

McCain -- seeking a fifth term as senator -- was ahead 58.8 percent to 29.8 percent, with 20 percent of precincts reporting, according to The Associated Press.

The race between McCain -- the 2008 GOP presidential nominee -- and conservative talk show host Hayworth started off nasty and didn't get any friendlier down the home stretch. Forced to spend $20 million in the campaign, McCain was driven to the right on some issues as Hayworth accused him of not being a true conservative.

McCain is expected to easily beat any one of the four Democratic primary candidates in the solidly red state.

In Florida -- one of four other states to hold primaries Tuesday -- millionaire political newcomer Rick Scott claimed victory over Florida Attorney General Bill McCollum in the Republican primary for governor.

In a speech to supporters, Scott, who spent $50 million of his own fortune since joining the race in April, alluded to the divisive nature of his fight against McCollum, the party-establishment favorite and former Congressman.

"Some of you may have noticed this was a hard-fought race. We talked a lot about our differences, but tonight it's time to remember those things that bring us together -- to recall our core beliefs and recommit ourselves to fighting for our principles," Scott said. "The Republican Party will come together, and the reason we will come together is our shared devotion to the values that make America great."

Scott was ahead of McCollum, 46.5 percent to 43.4 percent, with 96 percent of precincts reporting, according to The Associated Press.

The winner will face a November general election against Alex Sink, Florida's chief financial officer, whom CNN projected will win the state's Democratic primary for governor.

In a different race featuring a political veteran against a self-funded candidate with deep pockets, Rep. Kendrick Meek declared victory over billionaire Jeff Greene in Florida's Democratic primary for U.S. Senate.

Meek led Greene 57.3 percent to 31.2 percent, with 97 percent of precincts reporting, according to The Associated Press.

Meek will take on Marco Rubio -- who won the Republican primary for Senate -- and Gov. Charlie Crist, a Republican-turned independent, in the general election.

In an e-mail Tuesday night, Meek thanked supporters for lifting him past Greene, a billionaire real estate investor who was funding his own bid after making a fortune betting against the housing market.

"The pundits thought this seat could be bought. Our critics wrote us off. But together, you and I proved them wrong," Meek's message said.

Meek used his victory speech minutes later to try and establish himself as the candidate for Democratic voters in November. He noted that he is "running against two conservative candidates" -- a clear swipe at Crist, who could siphon votes from Meek in November. "I made the case, and I am the real Democrat in this race," Meek said.

Crist, who avoided a primary battle with Rubio by announcing an independent candidacy, framed the three-way Senate race as a choice between him, the hard right and the hard left.

"If ... you want somebody who wants to fight the gridlock in Washington and put the people first instead of the party, [and] do what's right for Florida rather than what's right for Washington or right for just Republicans or Democrats, then you have an alternative," he told CNN's "Larry King Live" on Tuesday night.

Rubio told his supporters that Crist and Meek would be the candidates for voters who already "like the direction that America is headed."

"If, on the other hand, you are unhappy with the direction that Washington is taking America ... then there is only one person running, there is only one campaign in Florida in 2010 that is offering to stand up to that agenda," Rubio said.

Florida's Democratic primary for Senate and the Republican primary for governor were two races in which two deep-pocketed political novices came virtually out of nowhere this spring, but managed to transform a pair of primaries into two of the most outlandish contests of the 2010 cycle.

Meek, a Democrat from Miami, Florida, had a clear path to his party's Senate nomination until April, when Greene decided to fund his own bid.

Both campaigns quickly trained fire on one another, with Meek dubbing Greene a "Meltdown Mogul." Greene linked Meek to an indicted Miami real estate developer and questioned Meek's commitment to Israel, a weighty charge in a state with a large population of Jewish voters.

Republicans also were grappling with a divisive primary in the governor's race between McCollum, a former Congressman backed by the state's party establishment, and Scott, a millionaire former health care executive.

Scott spent $50 million of his personal fortune since joining the race in April, mostly by blanketing Florida's expensive television and radio airwaves with advertisements questioning McCollum's conservative bona fides. He eventually stumbled on the campaign trail as he faced questions about his management of two health care companies that went on to face legal problems.

Arizona's GOP Senate race also was bitter, with Hayworth insisting that after running to the right in the primary, McCain would seek to build his legacy by cutting deals with President Obama and the Democrats if re-elected.

Fights over immigration fueled the race, as Hayworth and Tea Party activists challenged McCain's previous efforts at reform, which called for tougher border security, but included a pathway to citizenship for some undocumented workers.

But McCain defended Arizona's new immigration law, State Bill 1070, and went on a six-stop statewide tour with Arizona Gov. Jan Brewer, whose signing of the bill earned her wide praise from conservatives, and made her the symbol of opposition to amnesty.

On Tuesday night, McCain told supporters he was convinced that Republicans will regain majorities in both the Senate and the House.

"And when we do, we will stop the out of control spending and tax increases and repeal and replace Obamacare," McCain said. "We will keep families in their homes, we will create new jobs and we will allow our businesses to grow without Washington interference. We will secure our borders, defend our nation and bring our troops home from Afghanistan with honor and victory."

Other races:

-- Republican Gov. Jan Brewer of Arizona easily won her party's gubernatorial nomination Tuesday night, according to a CNN projection. Brewer captured 87 percent of the vote in the GOP primary election, according to early unofficial vote returns from the AP.

-- In Alaska, incumbent Republican Sen. Lisa Murkowski, with one of the best-known political names in the state, is pitted against an unknown challenger. But attorney Joe Miller's campaign picked up the support of the Tea Party movement and the backing of former Alaska Gov. Sarah Palin.

In a last-minute robocall for Miller, Palin went after Murkowski saying, "Lisa Murkowski has voted with the Democrats more than any Republican up for re-election this year. She waffled on the repeal of ObamaCare, co-sponsored cap and trade and voted for TARP." But some think that endorsement of Miller is at least partly personal. Palin had tangled with Murkowski's father Frank and defeated him in the 2006 governor's race.

The Tea Party made its support known again in the final hours, promoting the more than half-million dollars it put into Miller's campaign.

-- Also in Alaska, voters decide on gubernatorial nominees. Gov. Sean Parnell, who replaced Palin when she resigned last year, faces two challengers in the GOP primary. Two Democrats are battling for their party's nomination.

-- Longtime Republican Gov. Jim Douglas is retiring in Vermont, giving Democrats hope of competing for the seat. Five Democrats are running in what's considered a tight race. The winner will face GOP Lt. Gov. Brian Dubie.

-- In Oklahoma, two U.S. House Republican nominations will be decided in a runoff. In the 2nd District, Republicans think they have a chance for a pickup this fall against conservative Democrat Dan Boren. And two Republicans are battling to likely replace Rep. Mary Fallin, who's running for governor.

-- A 10-way Republican primary in the Arizona 3rd Congressional District race to replace retiring Rep. John Shadegg has attracted national attention because Ben Quayle, son of former Vice President Dan Quayle, is one of the candidates. The winner will face Democrat John Hulburd in the fall in the heavily red district.

CNN's Peter Hamby, Steve Brusk, Rachel Streitfeld, Jeff Simon and Mark Preston contributed to this report.