"I don't think that one party's going to be able to fix this by themselves," the South Carolina Republican said Saturday at a town hall event. "I think the President should reach out to Democrats, I should reach out to Democrats, and we should say, 'Let's take a shot at doing this together because it ain't working doing it by ourselves.'"
After Republicans were forced to pull their bill to replace Obamacare from the floor of the GOP-controlled House on Friday, Trump blamed Democrats and vowed to let Obamacare "explode."
"We had no Democrat support. We had no votes from the Democrats," Trump said. "They weren't going to give us a single vote, so it's a very, very difficult thing to do. I think the losers are (House Minority Leader) Nancy Pelosi and (Senate Minority Leader) Chuck Schumer because now they own Obamacare. 100% own it."
Graham has been a frequent critic of the Affordable Care Act and reinforced that Saturday.
"Here's what I think about health care: Obamacare is a disaster and it's going to collapse," he said at the Columbia, South Carolina, event, drawing boos from the crowd.
But the long-serving senator said the Trump administration is going to have to learn to work with Democrats if it wants to implement its vision for this country.
"At my core, I'm a fiscal and social conservative, but here's what I believe -- I can't run the country by myself, and we have to work together," he said. "If you want to save this country from becoming Greece, you need Republicans and Democrats to work together to reform entitlement programs before it's too late."
Sen. Bob Corker, a member of the Budget Committee, made a similar point Friday.
"At some point, on behalf of the American people, we have to resolve the issues that are driving up costs, limiting choices, and causing the individual market to spiral downward," the Tennessee Republican said. "I stand ready to work with the administration and my colleagues on both sides of the aisle in order to fix our broken health care system."
Graham, who has long called for an investigation into Russia's involvement in the 2016 election, also said Saturday that politicians should not hamper that investigation.
"We've learned that the FBI is investigating Trump campaign operatives for potential ties to Russia," he said. "Here's my belief: It goes where it goes. No politicians should stand in the way. We should let the FBI do their job. And what happens happens."
On Friday, House Intelligence Committee member Jim Himes accused California Rep. Devin Nunes of persistently serving "the interests of Donald Trump" as chairman of the committee, which is probing alleged ties between Trump associates and Russia.
"Devin, as much as I appreciate him and consider him a friend, has demonstrated on multiple occasions that he often serves the interests of Donald Trump," the Connecticut Democrat told CNN's Alisyn Camerota on "New Day." "Once again, we were shown why this should be done by an outside commission."
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Lindsey Graham on health care: Republicans and Democrats need to work together - CNN International