HOUSTON—A split between Texas Gov. Greg Abbott and Houston Mayor Sylvester Turner over whether the metropolis should have been evacuated is raising questions about officials’ response to damaging floodwaters as a catastrophe continues to engulf the region.

Mr. Turner, a Democrat, and other local officials urged residents to stay in their homes as Hurricane Harvey, which has since downgraded to a tropical storm, approached Houston on Friday.

But at a Friday news conference, Gov. Abbott, a Republican, suggested otherwise. “Even if an evacuation order hasn’t been issued by your local official, if you’re in an area between Corpus Christi and Houston, you need to strongly consider evacuating.”

By Sunday, the storm had poured as much as 24 inches of rain in 24 hours onto areas that had been soaked the previous day. Five fatalities have been reported in the Houston area, according to the National Weather Service, though Mr. Turner said Sunday that only one in the city was confirmed so far as storm-related. More than 3,000 water rescues had been performed by Sunday afternoon, officials said.

The offices of Messrs. Turner and Abbott didn’t immediately respond to requests for comment.

Emergency workers are trying to help those stranded in flooded areas of Texas as rain from what remains of Hurricane Harvey continues to batter the coast. Drone footage shows the extent of the damage across communities. Photo: AP

Families were stranded on rooftops, and parts of the city that residents say hadn’t flooded before were submerged.

The National Oceanic and Atmospheric Administration’s Weather Prediction Center is now projecting that some areas could see as much as 50 inches of rain—around as much as the region gets in a year—up from an earlier estimate of up to 40 inches.

Looming over the decision not to evacuate was Houston’s experience with Hurricane Rita. More than 100 people died while evacuating as 2.5 million people fled that storm in 2005. Some of the stories were horrific—23 nursing home patients were killed as a bus evacuating them caught fire and exploded near Dallas. Houston hasn’t ordered evacuations ahead of hurricanes since then.

Retired Army Lt. Gen. Russel Honorè, who led the Department of Defense response to hurricanes Katrina and Rita, said that experience has made officials wary of ordering an evacuation. But, he said, officials should have evacuated residents of flood-prone neighborhoods as well as other vulnerable populations like the elderly and homeless.

“I’m not trying to be critical of the mayor and history will prove whether they guessed right or they guessed wrong,” he said. “But I do not believe we should leave people in [a] place we know is going to flood. It’s counterintuitive.”

Video of Hurricane Harvey from International Space Station shows showing its scale, as authorities worry about ongoing flooding from the massive storm. Photo: NASA

He also said Gov. Abbott should immediately mobilize the entire National Guard. So far, 3,000 guardsmen have been activated. Mr. Honorè said the state would need closer to 15,000. “I’m very concerned because they don’t understand the scale they need for the response,” he said.

Mr. Turner defended his decision on Sunday, saying it would have been foolish to evacuate 6.5 million people from Houston and surrounding areas without knowing the course of the storm.

“There is no question in my mind, the best decision and the right decision was to tell people in Houston and Harris County don’t get on the road,” Mr. Turner said in a television interview.

“Now is not the time to second guess the decisions that were made,” Mr. Abbot said Sunday. He said he left several messages on Mr. Turner’s cellphone offering assistance, but hadn’t heard back.

Harris County Judge Ed Emmett, the area’s top emergency official and a Republican, bristled at criticism that the city should have evacuated.

“To suggest we should have evacuated two million people is an outrageous statement,” he said. “What we’re facing now is an effort to respond to a tragedy.…We’ve never seen water like this before.”

Gov. Abbott said at a press conference that there was good communication between state and local officials. “I have spoken with County Judge Emmett on a daily basis…to offer help the state of Texas can provide. We have moved beyond whether or not there should have been an evacuation and we are at the stage where we need to respond to the emergencies and necessities the people of Houston have.”

R. David Paulison, the former acting-director of the Federal Emergency Management Agency appointed by President George W. Bush after Hurricane Katrina, said the decision to call for an evacuation is among the most difficult.

Mr. Paulison said during Hurricane Katrina, FEMA urged Ray Nagin, the mayor of New Orleans at the time, to evacuate and was dismayed he didn’t. But during Katrina, the path of the storm was clear, Mr. Paulison said, which hasn’t been true for Harvey.

“Its tough to evacuate a big city like that,” he said. “I’m reluctant to second guess anyone at this point.”

At Houston’s George R. Brown Convention Center, which has been turned into a giant shelter, dozens of bedraggled people departed Metro buses and walked inside Sunday afternoon. Some had harrowing stories of escape, but were reluctant to criticize city officials.

Michael Williams, 36 years old, got there after the Metro bus he was riding got stuck in high water on Interstate 10 Saturday night, requiring all 20 passengers to be rescued. They spent the night at a Metro bus station before being brought to the convention center.

Mr. Williams, who works cleaning Minute Maid Park, the Houston Astros’ baseball stadium, had been trying to make a last-minute Wal-Mart run when he was caught in the waters. He said he had no criticism of officials for not calling a mandatory evacuation, saying flash flooding was a part of life in Houston.

“Houston floods fast. They’re doing all this construction but we still got no [storm] outlets,” he said. “But I ain’t mad. In a situation like this, you know the worst might happen.”

—Miguel Bustillo

contributed to this article.

Write to Christopher M. Matthews at christopher.matthews@wsj.com