Senator Robert Menendez of New Jersey maintained his innocence on Tuesday and vowed to appeal his conviction on sweeping charges of bribery, extortion and obstruction of justice.

But the weeks and months ahead are likely to be perilous, and his conviction raises urgent questions about whether he remains a senator or a political candidate this fall.

Mr. Menendez offered few immediate answers.

Judge Sidney H. Stein set a hearing to sentence Mr. Menendez and his co-defendants for Oct. 29, just a week before Election Day. The senator, who was released on his own recognizance, could face up to 20 years in prison.

It took only a matter of minutes, though, for Mr. Menendez, a Democrat, to face renewed calls to resign his Senate seat, including from Senator Chuck Schumer of New York, the majority leader, and Gov. Philip D. Murphy of New Jersey, who would appoint a replacement.

The Constitution does not prohibit felons from serving in Congress or running for re-election. And some of Mr. Menendez’s longtime allies said before the verdict that they expected him to dig in for a fight.

That path could lead to peril. If Mr. Menendez refuses to go voluntarily, he could face the threat of a rare expulsion vote in the Senate.

His prospects of holding his seat as a political independent seemed similarly vexed. Democrats already nominated a candidate, Representative Andy Kim, to replace him.

Mr. Menendez had said before the trial that he might run as an independent if acquitted. It was not immediately clear whether he would follow through. While there is next to no chance he could win, he could potentially siphon away crucial votes from Mr. Kim in a close contest.