DALLAS, Texas (CNN) -- Arizona Sen. John McCain, whose White House aspirations went into a nose dive last summer, clinched the Republican Party's presidential nomination Tuesday night with a sweep of GOP contests in four states.

"I am very, very grateful and pleased to note that tonight, my friends, we have won enough delegates to claim with confidence, humility and a great sense of responsibility, that I will be the Republican nominee for president of the United States," McCain told supporters in Texas.

CNN estimates that McCain has amassed 1,195 delegates to the GOP's September convention in Minneapolis-St. Paul, Minnesota, four more than the 1,191 needed to claim the party's nomination.

"Now, we begin the most important part of our campaign: to make a respectful, determined and convincing case to the American people that our campaign and my election as president, given the alternative presented by our friends in the other party, is in the best interest in the country that we love," McCain said.

"The big battle's to come," he said. "I do not underestimate the significance nor the size of the challenge." Watch McCain address supporters after sweeping Tuesday's contests »

McCain's last leading rival, former Arkansas Gov. Mike Huckabee, bowed out of the race after his projected losses in Texas, Ohio, Rhode Island and Vermont and urged his supporters to back the Arizona senator in November.

"It's now important that we turn our attention not to what could have been or what we wanted to have been but now what must be, and that is a united party," Huckabee said. Watch as Huckabee ends his presidential bid »

Claiming the title of presumptive nominee will give McCain a head start on the general election campaign while Democratic contenders Hillary Clinton and Barack Obama are still locked in a battle for their party's title, said Alex Castellanos, a GOP strategist and CNN contributor. Allocate delegates yourself and see how the numbers add up »

"Tomorrow, he can get started," Castellanos said. "He'll have the [Republican National Committee] behind him. He'll have a broad base of financial support. It's a big step. Meanwhile, it looks like the Democrats are engaged in the land war across Russia, so he's got a big advantage now."

Both Clinton, the New York senator and former first lady, and Obama, the first-term senator from Illinois, called McCain on Tuesday night, campaign officials said. Obama told McCain he looks forward to running against him in the fall, campaign spokeswoman Jennifer Psaki said.

McCain is slated to go to the White House on Wednesday to receive the endorsement of President Bush, according to two Republican sources.

The Arizona senator's campaign -- his second run for the White House -- was largely written off last summer amid outspoken opposition from the party's conservative base, a major staff shakeup and disappointing fundraising. But the former Navy pilot and Vietnam prisoner of war rebounded with wins in January's primaries in New Hampshire and South Carolina, the state where his first presidential bid foundered.

"There were times, obviously, when my political campaign was not viewed as the most viable in America, as you probably know," he told reporters in San Antonio earlier Tuesday. "In fact, I was reminded of the words of Chairman Mao, who said it's always darkest before it's totally black."

McCain's fortunes also rebounded as U.S. commanders in Iraq credited the 2007 launch of a campaign to pacify Baghdad and its surrounding provinces with a sharp decline in American and Iraqi casualties. The senator had been one of the most outspoken advocates of the shift and has blasted his potential Democratic rivals for calling for the withdrawal of U.S. combat troops from the widely unpopular war.

"This is a man with a lot of trials in his life," said former Education Secretary William Bennett, a CNN contributor. "He's had a lot of downs; he's been up, and this is a big up."

McCain has been turning his fire on the Democrats, for whom Tuesday's races in Ohio and Texas are seen as pivotal. See scenes from Tuesday's voting »

But Democrats have been pounding McCain over his January comment that he would be satisfied if U.S. troops remained in Iraq for 100 years, as long as the insurgency there died down. And Democratic National Committee Chairman Howard Dean has attacked his reputation as a reformer over the past week, accusing McCain of trying to evade federal spending limits by opting out of public financing after using the promise of federal funds to obtain a bank loan and automatic ballot access for his primary campaign.

Dean told CNN on Tuesday that McCain "really is the focus of what we're doing now, in terms of his ethics problems and his problems with the war and his problems with the huge deficits that they've run up on the Republican side."

In 2000, McCain upset then-Texas Gov. George W. Bush in the New Hampshire primary by touting "straight talk" and his record as a Republican maverick. Bush came back in South Carolina amid a divisive and bitter campaign that left McCain denouncing leaders of the party's religious conservative wing as "agents of intolerance," and Bush went on to win the presidency.

Since then, McCain has enraged conservative leaders by opposing Bush's signature tax cuts, co-sponsoring the campaign finance reform law that now bears his name and supporting a controversial White House-backed plan to offer a path to citizenship for illegal immigrants. But their support was spread among a fractured GOP field, and their main standard-bearer, former Massachusetts Gov. Mitt Romney, quit the race after a disappointing showing in February's Super Tuesday primaries.

Exit polls in Texas and Ohio found that about three-quarters of Republicans would be satisfied with McCain as their nominee, however. Those surveys found that the economy was the top issue for GOP voters in both states -- and by a wide margin in Ohio, which has seen a sharp decline in manufacturing jobs in the past decade.

Although national security issues are a strong suit for McCain, Castellanos said he might need some help if a weakening economy is the central issue in November.

"It's never been Sen. McCain's strength," Castellanos said. He said McCain would need to make the case that "I'm going to grow this economy; Barack or Hillary, they're going to grow government."

McCain had amassed 1,047 delegates before Tuesday, according to CNN estimates. At stake in Tuesday's contests were 256 delegates, allocated on a winner-take-all basis by statewide or congressional district results. E-mail to a friend

CNN correspondent Dana Bash and political editor Mark Preston contributed to this report.

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