With Baylor University at war with itself over the firing of its football coach in the wake of a sexual-assault scandal, this qualifies as good news: The NCAA is not planning to bring the hammer down on Baylor the way it did Penn State.

The National Collegiate Athletic Association has notified Baylor that it won’t exert its executive authority to impose sweeping sanctions against the school for broad institutional failings, and will instead follow its normal investigative process, according to people familiar with the matter.

The program still faces a more narrow NCAA probe, even as it copes with the increasingly messy aftermath of its decision in May to fire popular football coach Art Briles.

Briles and university president Kenneth Starr were dismissed after an external investigation cited a litany of failures in responding to sexual violence on campus, particularly involving the football program. University officials offered few specifics at the time. But last month, The Wall Street Journal reported that the investigation uncovered 17 women who reported sexual or domestic assaults involving 19 players, including four alleged gang rapes, since 2011.

The NCAA decision indicates Baylor—at least for now—will not be subject to the sort of harsh sanctions the NCAA imposed on Penn State in the wake of the Jerry Sandusky child-molestation scandal. That case, in which the former assistant football coach was convicted of abusing boys over a 15-year period, was viewed by many as a potential precedent for Baylor since it also involved alleged criminal activity within the athletic department but not clear-cut NCAA violations.

Under pressure from the NCAA, which cited a broad “failure of institutional integrity,” Penn State after the Sandusky scandal vacated 111 wins under Coach Joe Paterno, accepted a four-year bowl ban, reduced football scholarships and agreed to pay $60 million to fight child abuse. Many of those sanctions have since been vacated after a series of legal challenges.

The NCAA’s aggression in the Penn State case “really backfired,” says B. David Ridpath, an Ohio University professor of sports administration. “I doubt you will ever see the NCAA do anything like that again.”

Prior to the TCU-Baylor game in Waco on Nov. 5, Jon McClellan of Houston surfs through a mound of shirts with #CAB printed on them, standing for Coach Art Briles. Photo: Tony Gutierrez/Associated Press

Representatives for the NCAA and Baylor declined to comment.

Baylor still faces major challenges, however, with both the NCAA and a campus community that has been deeply divided in the aftermath of the scandal.

Baylor, based in Waco, Texas, has said it is cooperating with the NCAA, and people familiar with the matter say that probe will likely continue for months. That investigation will likely focus more narrowly on whether Baylor athletes received preferential treatment through the school’s disciplinary process, which could be an impermissible benefit under NCAA rules, the people said.

But that probe may be the least of Baylor’s worries, as the scandal’s aftermath at the nation’s largest private Baptist university has grown messier by the week.

On Thursday, a group of alumni—including some of the school’s biggest donors—are hosting a rally in Waco for a new group they have founded called Bears for Leadership Reform, in a highly unusual effort to change the governance of a major university.

Meanwhile, the football team, which won its first six games and was ranked as high as No. 8, has lost its last two games, including a 62-22 shellacking by TCU on Saturday. The night before the TCU game, the team’s assistant coaches signed a Twitter message defending Briles, which seemed to further galvanize alumni who feel the coach has been unfairly scapegoated for what Baylor officials have acknowledged is a campus-wide problem.

Some fans sold shirts in the parking lots with the initial CAB—for Coach Art Briles—and the occupants of one luxury box flew a black flag with the same logo. Bears for Leadership Reform is led by alumni including Drayton McLane, the billionaire businessman whose name is on the Baylor football stadium; John Eddie Williams, a Houston trial lawyer whose name is on the field itself; and former Texas Gov. Mark White.

Williams said the new group is committed to improving campus safety and transparency around how the school is tackling the issue of sexual violence.

He said the group is not asking for Briles to be brought back, but questioned why no members of the board of regents have accepted responsibility for their role in the scandal. “How could this number of assaults have occurred and the board not know?” Williams said.

Williams would not say whether his plans to give further donations to Baylor have changed, saying only that he is “gravely concerned” about the leadership of the school.

Baylor officials have said they are braced to deal with such fallout.

“There will always be problems with fat-cat alumni saying we are not going to give you millions of dollars unless you do x, y and z. You have to be strong enough to say, I guess you are not going to give us millions of dollars,” said Baylor regent J. Cary Gray in an interview with the Journal last month.

Gray and the other regents who spoke to the Journal said they were upset to learn that Baylor hadn’t been more caring to victims of sexual assault, and pledged to do better. Gray said the regents want to “align all of athletics, not just football, with the rest of the university and its mission.”

Write to Brad Reagan at Brad.Reagan@wsj.com