Mitt Romney's campaign team has appealed for his rivals to quit the Republican presidential race, claiming that it is near impossible for them to catch him after the Super Tuesday results and that by staying in they are boosting President Barack Obama's re-election chances.

But Rick Santorum, boosted by his three wins on Tuesday and running Romney close in Ohio, was out on the campaign trail again on Wednesday and stubbornly refusing to concede, intent on pushing Romney to the end. The former House Speaker, Newt Gingrich, too was out campaigning and refusing to pull out.

Another series of bruising and bitter contests beckons, with the next big test in Illinois on 20 March, which will again pit Romney against Santorum and expose Republican faultlines.

Despite his advantages in dollars and organisation, Romney failed to make a decisive break on Super Tuesday, unable to put together a winning coalition of voters that would knock out his opponents and concentrate on gathering money and energy in preparing to take on Obama for the White House in November.

Democrats expressed glee at the outcome. David Axelrod, Obama's senior adviser, in a phone conference with reporters from campaign headquarters in Chicago, described the Republican campaign as a death march. "Romney wanted to put the campaign away. Instead of Super Tuesday, it became super glue today. They are still stuck."

Jim Messina, Obama's campaign chief, added: "Every primary we have seen shows how disappointed Republicans are with their candidates … dissatisfaction on Super Tuesday was super-sized."

The Romney campaign hoped Tuesday's votes would have proved to be more decisive and had been planning a more robust appeal to Santorum and Gingrich to quit. Instead, they had to opt for a more muted plea.

One of Romney's campaign advisers, Rich Beeson, argued that the maths showed Santorum and Gingrich were unlikely to catch up and the only beneficiary of a protracted Republican race would be Obama.

"Super Tuesday dramatically reduced the likelihood that any of governor Romney's opponents can obtain the Republican nomination. As governor Romney's opponents attempt to ignore the basic principles of math, the only person's odds of winning they are increasing are President Obama's," Beeson said.

With about 20 out of the 50 states having voted in Republican primaries and caucuses, Romney has accumulated only about a third of the 1,144 delegates to the Republican convention in August that he needs to secure the nomination. The race seems set to rumble on through the spring and summer, culminating in Utah in June.

Santorum, after his election night party in Ohio on Tuesday, began campaigning on Wednesday in Kansas, which votes on Saturday and which polls suggest he stands a good chance of winning, and in Mississippi, which votes next Tuesday and looks favourable either for him or Gingrich. Alabama, also up next Tuesday, could go to either Santorum or Gingrich.

Brad Coker, head of southern-based Mason-Dixon polling, said the Republicans should calm down. "They are freaking out because it is March and the contests are still going on. Historically, it is rare for these fights to be over quickly," he said.

He said Romney would be the nominee. While upcoming states did not look favourable to him the big states that followed – California, New York and New Jersey – would all go for Romney, as well as smaller states such as Connecticut and Rhode Island. He might also win Illinois.

Romney will be favourite in Illinois among better-off voters in urban Chicago while Santorum is likely to win the backing of the Republican working-class, social conservatives, Christian evangelicals and rural areas.

Super Tuesday, the biggest night of a campaign that began on 3 January, resulted in Romney taking six of the 10 states at stake: Vermont, Virginia – where Santorum and Gingrich were not on the ballot because of organisational incompetence – Massachusetts, Alaska and, the big prize of the night, Ohio.

"Romney didn't just win Super Tuesday – he crushed it," his press spokesman Eric Fehrnstrom tweeted. Another Romney campaign adviser said that Super Tuesday had been the last opportunity for Santorum and Gingrich to close the delegate gap and they had failed to take the opportunity.

But the Santorum and Gingrich camps, not unsuprisingly, had a different take, saying the results again showed up Romney's vulnerabilities and the unhappiness with him among conservatives.

Santorum took Oklahoma, Tennessee and North Dakota, while Gingrich, adding to the confused picture, won his home state, Georgia. This was despite being outspent by Romney in advertising.

In the showdown in Ohio, traditionally one of America's big swing states, Romney just held off a strong challenge from Santorum, taking 38% of the vote to Santorum's 37%, with Gingrich trailing in third place on 15% and Ron Paul, fighting a modest campaign, on 9%.

Romney hinted at the toll the long campaign is taking on him as he flew from Ohio to Boston on Tuesday night. "I'm looking forward to being home, sleeping in our own bed tonight. This has been a long one," Romney said.

Adding in Super Tuesday, Romney has now won 13 states, Santorum six –he is almost certain to add Missouri later this month, having come out on top in a test-run last month – and Gingrich, two. Paul has yet to record a win and had hoped to take one of the smaller states: Idaho, North Dakota or Alaska. He admitted his chances of winning the nomination are slim.

According to the Associated Press, Romney added about 200 delegates to his tally on Super Tuesday, bringing him total to 415. Santorum has 176, Gingrich, 105 and Paul, 47.

Exit polls showed voters split between those who regarded Romney as the more electable in a one-to-one with Obama and those conservatives who do not view Romney as a "a true conservative" or having "a strong moral character.