Federal district courts in Florida and Virginia have now ruled the law is unconstitutional, although the issue is widely expected to land in the U.S. Supreme Court given that two other courts have upheld the law.

Still, McConnell said it would be a "dereliction of duty" for Republicans not to fight for repeal. He also said the law failed to deliver on key Democratic promises, such as lower health insurance premiums, and reiterated that the bill is hurting U.S. job creation, not helping it.

Later in the morning, Sen. Orrin HatchOrrin Grant HatchOvernight Finance: NAFTA defenders dig in | Tech pushes Treasury to fight EU on taxes | AT&T faces setback in merger trial | Dems make new case against Trump tax law | Trump fuels fight over gas tax What sort of senator will Mitt Romney be? Not a backbencher, even day one Lawmaker interest in NAFTA intensifies amid Trump moves MORE (R-Utah) called explicitly for a vote in the Senate, but said Democrats are likely to worried about growing opposition to the law to call the vote.

"Ultimately you have to ask why the Democratic majority would deny us this vote," Hatch said. "I think I know the answer. It has a great deal to do with members of the [Democratic] Caucus who know their constituents hate this law."

In his opening remarks this morning, Senate Majority Leader Harry ReidHarry Mason ReidWATCH: There is no Trump-Russia collusion and the media should stop pushing this The demise of debate in Congress ‘North by Northwest,’ the Carter Page remake MORE (D-Nev.) gave no indication that he would schedule a vote on healthcare repeal, and instead defended the law.

"The health reform bill has already saved lives, and saved lots of money," Reid said, arguing that children are better covered and older people can buy more affordable drugs under the law. He also said the law helps crack down on Medicare fraud, and that lower business taxes make it easier for companies to provide health insurance to employers.

"As more parts of the law go into effect it'll go even more good," he said. "Even more lives and more money will be saved."

"Lawsuits and lawmakers' efforts to repeal this bill is nothing more than an attempt to raise taxes on small business, add more than a trillion and a half dollars to the deficit, force seniors to pay more for their prescriptions, and let insurance companies once again stand in the way of a child and the medical care he or she needs."

Updated at 11:40 a.m.