"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.
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.
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