There are unspoken agreements within a party to let the majority leader — in concert with the relevant committee chairmen — decide how bills will be managed on the floor. Procedural stunts against a fellow party member are frowned upon. “I’m not going to tell you how I could advise another senator,” said Senator John Cornyn, Republican of Texas and the majority whip, adding that he could not surmise what Mr. Cotton was thinking in his strategy.

For Mr. Rubio, whom Mr. Cotton represented in his move to force an Israel vote, the stakes are relatively low. Mr. Rubio is leaving the Senate to run for the Republican presidential nomination and needs to curry favor with primary voters, not members of the Senate.

Mr. Cotton, who declined to comment Tuesday, has often operated as a caucus of one during his first four months in the Senate. As Mr. Corker worked on a bipartisan measure that would force the White House to seek congressional approval to roll back certain sanctions against Iran in the event of a nuclear deal, Mr. Cotton chose to circulate among his Republican colleagues a combative letter to Iranian leaders suggesting that Congress could undo any nuclear deal. That letter was at least a temporary success as 46 colleagues agreed to sign it, though not Mr. Corker.

Republicans — and many Democrats — have enjoyed the ability to attach amendments to bills in the Republican-controlled Congress, a luxury Mr. Reid generally did not allow. For example, during a vote this year to approve the Keystone XL oil pipeline, senators offered scores of amendments, carefully managed by members of both parties, said Senator John Hoeven, Republican of North Dakota.

But Mr. Cotton used a procedural sleight of hand to essentially jump ahead of his colleagues without warning. That upset Republicans and alienated Democrats, who had been going along with the process to get the bill off the floor. They already did not relish the idea of having to vote against things like an amendment to require the release of captive Americans, which sounded great on paper but would have caused the president to veto the bill if passed.

Two senior senators suggested Mr. Cotton’s arcane procedural moves could be used against him on the floor at some point, and they were watching to see if he proceeds as aggressively in the future.

However, some colleagues said Mr. Cotton and Mr. Rubio were well within their rights to use procedure to their advantage. “We’re all going to do what we think is right to make a bill better,” said Senator Cory Gardner, Republican of Colorado.

Mr. Corker, whose own amendments now seemed doomed, was sanguine Tuesday. “Around here, you kind of take it a day at a time,” he said. As for Mr. Cotton, he said, “There hasn’t been a harsh word between us.”