McConnell said in a statement that the Senate "will continue (their) work on legislative items and nominations and will defer consideration of the Better Care Act".
Senate Republicans are struggling to make good on a almost decade-long promise to repeal and replace Obamacare, also known as the Affordable Care Act, and polls show their legislation is unpopular with voters.
She said on This Week that she is also concerned about the bill's giving insurers the option of offering bare-bones health plans. John McCain announced he would be home all week recovering from surgery.More news: Chances increase for Northern Lights across MI tonight
Two Republican senators - Susan Collins of ME and Rand Paul of Kentucky - have already said they will not support a motion to proceed to floor debate on the legislation.
Republican senators Susan Collins of ME and Rand Paul of Kentucky are against it, and others have expressed concern.
"You know, I don't think right now he does", Paul said. The bill rests on a knife's edge, and Wisconsin's Ron Johnson says his "yes" vote is in jeopardy. The 80-year-old Senate veteran was advised by doctors to remain in Arizona next week, his office said.
But Democrats, who are powerless to stop the bill unless more Republicans defect, are gleefully seizing on the opportunity the delay provides to hammer their GOP colleagues for not holding a single public hearing on the massive legislation.
The release detailed information regarding McCain's status and how the surgery went.More news: United States denies visas to Russian replacements for expelled diplomats - Moscow
Fellow Arizona Republican Sen.
"This bill actually I think has gotten much better as a result of the discussions we've had amongst ourselves, and I think it's something that once we agree to that we can sell to the American people as a better choice than the failures of Obamacare", he added.
The healthcare bill was already on rocky ground even with McCain, as many speculated whether McConnell could pull together 50 votes.
But McCain and his doctors have not revealed the cause.
Until Lee is satisfied the bill does what he wants, he seems to be making it clear he won't commit to supporting it.More news: Dow hits another record as stocks close higher
Sen. Jeff Flake, also from Arizona, put out a statement, saying he and his wife were praying for a speedy recovery.