But the gravest fears of Alabama’s critics were not realized. Some of that reflected years of advocacy by voting rights groups, including the concerted pushback that led the state, in 2015, to back away from a plan to close 31 driver’s license offices in rural areas, many of them predominantly black.

Alabama’s secretary of state, John H. Merrill, said allegations of irregularities, from the left and right, in the Dec. 12 election were not borne out. “It’s just people making things up, and they think they can because of what they’ve observed that happened before or what our history has been,” said Mr. Merrill, a Republican.

Still unclear is what role voting restrictions, including voter ID, are playing on turnout here and elsewhere. Exit polls are preliminary, but the ones available in Alabama suggest the share of blacks who cast ballots — roughly 41 percent of the African-Americans voters — exceeding the 35 or so percent of whites who turned out. The divide likely reflects a robust black turnout and modest participation from whites who were unenthusiastic about Mr. Moore, whose already-controversial candidacy was dogged by accusations of sexual misconduct.

One recent academic study concluded that the historic turnout gap between white and minority voters increased sharply — as much as fivefold — in states with the strictest voter ID laws, producing a “clear partisan distortion” favoring Republicans.

In Texas, where federal courts have invalidated parts of one of the nation’s toughest ID laws, a detailed analysis concluded that 3.6 percent of white registered voters in Texas lacked any legally acceptable ID — and 5.7 percent of Hispanic voters, and 7.5 percent of African-Americans. But among more likely voters who cast ballots in the 2010 and 2012 elections, only 1.4 percent lacked a valid ID. An estimated 600,000 registered voters lacked a photo ID that qualified them to vote under the law.