U.S. Senator Lindsey Graham (R-SC) and Senator Chuck Schumer (D-NY) (R) speak during a push for new bipartisan media shield legislation during a news conference at the U.S. Capitol in Washington, July 17, 2013. Also pictured is co-sponsor Senator Amy Klobuchar (D-MN) (L). REUTERS/Jonathan Ernst (UNITED STATES - Tags: POLITICS MEDIA)
Sen. Lindsey Graham (R-S.C.)  REUTERS/Jonathan Ernst

The fight over the president's health-care law should be saved for another day, Sen. Lindsey Graham (R-S.C.) said Sunday on ABC News's "This Week."

"To my colleagues in the House, on both sides and to my friends in the Senate, we're ruining both institutions," Graham said. "So, it is unrealistic to expect us to defund or delay Obamacare by shutting the government down."

That doesn't mean the fight against the health-care law should be abandoned, he said, adding that the measure will become a liability for the Democratic party.

"To our Democratic friends, you own Obamacare and it's going to be the political gift that keeps on giving," he said. "So the shutdown will be old news next year; Obamacare's faults will be front and center in 2014, if we don't screw this up."

And Graham wasn't alone. Sen. Kelly Ayotte (R-N.H.) echoed his sentiment on CBS's "Face the Nation."

"I believe that the defunding strategy was a failed strategy from the beginning," she said.