"I give us the best chance to unify," the Florida senator said Sunday in an interview with CNN's Jake Tapper on "State of the Union."
"Who do the Democrats fear most? Who do they not want to run against? I think everyone now acknowledges that's me," he said. "We've got to bring the Republican Party together. We're not gonna win -- we're not going to beat Hillary Clinton or Bernie Sanders -- if we're still divided in September or October."
While Donald Trump won South Carolina's primary Saturday, Rubio was buoyed by a slim second-place finish over Texas Sen. Ted Cruz -- and perhaps even more by a key rival, former Florida Gov. Jeb Bush, deciding to drop out after that primary.
He said the race has a "different dynamic now," with three real contenders -- Trump, Cruz and himself.
While no candidate has ever won both New Hampshire and South Carolina, as Trump just did, without winning the nomination, Rubio argued that GOP history isn't relevant because of this year's larger field of candidates.
"There's never been a race like this," he said. "You cannot apply the rules of the other races to this one. I think last night was truly the beginning of the real Republican primary."
There's another candidate who threatens to pick up some of Bush's support: Ohio Gov. John Kasich, who finished in second place in New Hampshire.
Rubio took several subtle shots at Kasich on Sunday, casting him as "primarily focused on Michigan" and suggesting Kasich doesn't have a serious national operation.
"You're down to three candidates who are running full-scale national campaigns," Rubio said, referring to himself, Trump and Cruz.
Rubio also offered a preview of how he'll attack Trump, touting his own conservative record -- including fighting the Florda's then-incumbent Republican Gov. Charlie Crist, for the 2010 Senate nomination -- against Trump's.
"I didn't just become a conservative two years ago, but I'm a conservative who can unite our party, grow our party," he said.
And as the campaigning heads to Nevada's contest on Tuesday's, Rubio scored the endorsement of Sen. Dean Heller
, who had previously backed Bush.
"I've become convinced that he's the candidate capable of uniting conservatives, growing our party and beating the Democrats in November," the Nevada Republican said. "And Marco understands Nevada, he grew up here. I am proud to offer my full support."