Robert F. Kennedy Jr., the vaccine critic who spawned a movement he calls “Make America Healthy Again,” moved one step closer to becoming the nation’s health secretary on Wednesday after the Senate voted along party lines to advance his nomination, setting the stage for a final confirmation vote, likely on Thursday.

The 53-47 vote was a stunning show of Republican support for President Trump, who embraced Mr. Kennedy, the scion of a liberal Democratic dynasty, and promised to let him “go wild” on health. And it was a sign of Mr. Kennedy’s strength; in the immediate aftermath of Mr. Trump’s election, many observers thought he would be unable to win confirmation.

Mr. Kennedy’s vocal criticism of vaccination made some Republicans deeply uneasy. During his confirmation hearings, he refused to accept the scientific consensus that vaccines do not cause autism, rejecting large scale studies that have found no link.

But at the same time, Mr. Kennedy declared himself in favor of the measles vaccine and the polio vaccine and vowed not to do anything as health secretary that would prevent or discourage people from taking those shots.

Among the most closely watched votes on Wednesday was that of Senator Mitch McConnell, the Republican leader and the Senate’s lone polio survivor, who voted in favor of moving ahead with the confirmation. That does not mean Mr. McConnell’s final vote is assured; he voted to advance the nomination of Tulsi Gabbard as director of National Intelligence, but voted against confirming her on Wednesday.

Mr. McConnell did not have a customary courtesy visit with Mr. Kennedy, and he had warned that “anyone seeking the Senate’s consent to serve in the incoming administration would do well to steer clear” of “efforts to undermine public confidence in proven cures.” Polio survivors were watching his vote closely.