WASHINGTON — The Senate will delay votes on a bill to dismantle the Affordable Care Act, the Senate majority leader, Mitch McConnell, announced Saturday night, because of a new obstacle to winning enough support for one of the Republican Party’s most cherished goals.

Mr. McConnell said the Senate would “defer consideration” of the bill, scheduled for this week, because Senator John McCain, Republican of Arizona, would be absent, recovering from surgery that he had on Friday to remove a blood clot above his left eye.

Mr. McConnell had said that he wanted to begin debate on the bill and pass it this week, using special fast-track procedures. But without Mr. McCain, Senate Republicans would not have the votes they need to take up or pass their bill to repeal and replace major provisions of the health care act that was the signature domestic achievement of President Barack Obama.

It was unclear how long the delay will be. “The leader has not announced a date” for the Senate to take up the legislation, said an aide to Mr. McConnell. “Just that we will defer.”