When Senator Robert Menendez of New Jersey goes on trial on federal corruption charges in less than three weeks, far more than his own fate hinges on the outcome.

If Mr. Menendez, a Democrat, is convicted and then expelled from the United States Senate by early January, his replacement would be picked by Gov. Chris Christie, the Republican governor of New Jersey and an ally of President Trump.

That scenario — where Mr. Menendez’s interim replacement would more than likely be a Republican — would have immediate and far-reaching implications: The Republicans would be gifted a crucial extra vote just as the party remains a single vote shy in the Senate of advancing its bill to dismantle President Obama’s signature health care law. Those potential consequences only heighten the drama around the first federal bribery charges leveled against a sitting senator in a generation.

“This one vote, this one vote — if he’s convicted or does a plea deal — could change the course of history on Obamacare. It’s remarkable,” said Steve Lonegan, a New Jersey Republican who unsuccessfully ran for Senate three years ago.