McCain's absence cast doubt on whether the Senate would be able to pass the legislation to dismantle and replace Obamacare. McConnell needs 50 "yes" votes for passage in a chamber the Republicans control by a 52-48 margin.
"While John is recovering, the Senate will continue our work on legislative items and nominations, and will defer consideration of the Better Care Act," McConnell said in a statement.
The announcement came after McCain's office said he would remain in Arizona next week following a procedure to remove a 2-inch (5-cm) blood clot from above his left eye.
Repealing and replacing President Barack Obama's signature healthcare law was a top campaign promise for President Donald Trump and Republicans in Congress.
Senate Majority Leader Mitch McConnell speaks at a Harden County Republican party fundraiser in Elizabethtown, Kentucky, U.S., June 30, 2017.Bryan Woolston
But two Republican senators have already declared their opposition to revised legislation unveiled on Thursday.
McCain has expressed concern about the healthcare bill but has not said how he would vote.
The 2008 Republican presidential nominee, McCain was resting comfortably at home in good condition after Friday's operation, his office said.
"There are few people tougher than my friend John McCain, and I know he’ll be back with us soon," McConnell said.
McCain's surgeons removed the clot during a minimally invasive craniotomy through an incision in the 80-year-old lawmaker's eyebrow. Tissue pathology reports would be available within the next several days.