"I give us the best chance ... to nominate a real conservative who can unite the party, grow the party... and ultimately defeat Hillary Clinton or Bernie Sanders," Rubio told CNN's Alisyn Camerota on Tuesday.
"The Democrats admit that I am the one they don't want to run against," he added.
Rubio finished in third place
in Monday's Iowa caucuses behind Texas Sen. Ted Cruz and billionaire businessman Donald Trump, a higher than expected result that has spotlighted his campaign.
But by staying near the top of the Republican presidential race, Rubio is hoping to unify the establishment
vote he needs to stay for the long run. He says as much when asked about the insults tossed at him by fellow candidates like former Gov. Jeb Bush
and Gov. Chris Christie
"I'm not running to beat up on other Republicans," Rubio said during an interview at a restaurant in New Hampshire. "Ultimately, I'm running to unify this party."
Rubio said he arrived in New Hampshire at 1:30 a.m. Tuesday, slept "four or five hours," and hit the campaign trail. New Hampshire has its presidential primary on February 9.
Camerota asked Rubio if he would take a different tone with voters in New Hampshire, which doesn't have Iowa's large population of Evangelical Christians.
"The campaign we launched in April last year ... will be the message I have in November this year. I'm not running two separate campaigns," he said. "America's greatest days are within our reach. But not if we stay on the road we're on right now."
On illegal immigration, Rubio said his White House plan would be to first focus on tightening border control -- not to have federal agents hunt down millions of families.
"I don't think the American people expect us to round up and deport 12 million people," he said.
Instead, Rubio proposes "finishing the wall" between the United States and Mexico, as well as improving the government's tracking of foreigners entering and leaving the country. Rubio said he would also enforce the federal government's E-Verify program,
extra paperwork that forces business owners to check the legal status of all new workers against Social Security and immigration records.
Rubio said those who break criminal laws should be deported immediately. But the vast majority of undocumented immigrants -- who aren't breaking criminal laws -- should be given a legal path to stay. Rubio suggests creating 10-year work permits for them.
"But I'm not gonna ram anything down anyone's throat, and we're not going to do it through unconstitutional executive orders like this president now," he said.