With 30 percent of precincts counted, The Associated Press declared that Houston’s Proposition 1 had gone down to defeat. The measure was taking a beating with 62 percent of voters against the measure and 38 percent in favor.

The vote capped a pitched 18-month battle between Houston pastors and Mayor Annise Parker, the first openly lesbian mayor of a major city, who had championed the ordinance allowing people to use restrooms, showers and other public accommodations based on their gender identity, not their biological sex.

In Kentucky, Republican businessman Matt Bevin defeated Democratic Attorney General Jack Conway by 53 to 44 percent, with independent Drew Curtis taking 4 percent in the race to succeed term-limited Democratic Gov. Steve Beshear.

Meanwhile, Mississippi Gov. Phil Bryant easily won a second term over Robert Gray, a Democrat who spent just $3,000 in a token campaign in the overwhelmingly Republican state.

Ohio voters overwhelmingly rejected Issue 3, which would have legalized recreational marijuana for adults 21 and over, handing the pro-pot movement its first defeat since Colorado and Washington approved legalization measures in 2012.

“When it comes to the broader debate about legalizing marijuana, the defeat of Issue 3 won’t be a case of ‘as Ohio goes, so goes the nation,’” said Tom Angell, chairman of Marijuana Majority, in a statement.

Unlike Ohio, the four states — Colorado, Washington, Oregon and Alaska — that have approved recreational pot for adults began with medical marijuana. “This was about a flawed measure and a campaign that didn’t represent what voters want,” Mr. Angell said.

Mr. Bevin, a Kentucky business owner who had challenged Senate Majority Leader Mitch McConnell in the 2014 GOP primary, ran as a political outsider with strong tea party support. He won the GOP gubernatorial primary in a crowded Republican field by just 83 votes.

In his victory speech, Mr. Bevin issued a clarion call for unity, praising Mr. Conway and saying “this is our opportunity to come together as one. We have much work ahead of us.” It is time “to get the overalls on, get the boots on and get out of bed,” he said.

His running mate — Lt. Gov.-elect Jenean Hampton, who introduced him Tuesday night — became the first black person to win a statewide race in Kentucky.

Mr. Conway made a concession and congratulations call before 9 p.m., telling supporters at the Frankfort Convention Center that he had told his rival that he “remained positive about moving this state forward, and that if he ever needed any assistance, that this Democrat was at his disposal.”