“He is showing passion, but what I am trying to do is build a party that has a bright future if we want one,” Graham said on CNN’s "Crossfire." “I can tell you this — the best thing the Republican Party can do is have a debate on ObamCare throughout 2014.”

Graham also said he should start calling the legislation “HillaryCare” after both Bill and Hillary ClintonHillary Diane Rodham ClintonSessions says FBI agent Peter Strzok no longer has his security clearance Melania Trump puzzles with 'I really don't care' jacket Grassley wants to subpoena Comey, Lynch after critical IG report MORE spoke about the law during a panel discussion at a Global Clinton Initiative event with Obama on Tuesday. 

“It is now HillaryCare,” he said. “I don’t know why the Clintons have adopted this.” 

Clinton is a potential presidential prospect in 2016. As the first lady in the 1990’s, she spearheaded her own failed attempt at a healthcare overhaul. 

Graham said he dislikes the healthcare law as much as Cruz, but he is not willing to see the government shutdown next Monday. 

While Cruz has launched an hours-long speech on the Senate floor, the Senate is still expected to vote Wednesday to advance the bill with final passage over the weekend. 

Before that time, Democrats are expected to vote to strip the bill of language defunding ObamaCare with a majority vote.  

Graham said the GOP may be able to fight for less significant amendments to the healthcare law, like repealing the medical device tax or temporarily delaying the mandate on individuals to buy insurance. 

“I am trying to get votes on a bill that I think can be dramatically changed if people on the Republican and Democratic side have an opportunity to actually talk about and vote on parts that we like,” he said.