Virgin Galactic Holdings Inc. won’t be allowed to operate space flights until aviation regulators finish investigating an earlier mission carrying billionaire entrepreneur Richard Branson, the Federal Aviation Administration said Thursday.

Virgin Galactic on Wednesday said the spacecraft, called the Unity, shifted out of its planned trajectory for one minute and 41 seconds as it returned to the ground on July 11, prompting the FAA to examine what happened. Three Virgin Galactic employees joined company founder Mr. Branson for the flight to the edge of space, which attracted extensive attention.

The company said it has been working closely with the FAA to support a thorough review of the flight and resolve the matter in a timely way. The company on Wednesday attributed the deviation in its flight path on July 11 to wind, and said pilots who commanded the ship responded appropriately to changing conditions as they brought the Unity back to the Spaceport America facility in New Mexico. Virgin Galactic also said then that the Unity never went outside of the geographic confines that had been established for the mission, and that those on board were never in any danger.

An FAA spokesman said Thursday the Unity ship can’t be operated until the agency approves a final report on the July 11 matter, or determines the issues related to the Unity’s shift out of its planned trajectory that day didn’t affect public safety.

The FAA spokesman declined to comment on how long the agency’s investigation would last. So-called mishap probes generally can be finished in a matter of weeks, he said, but more complex ones may take several months.

The agency’s examination of the July 11 flight may delay Virgin Galactic’s next expected space flight. The company said earlier Thursday it planned to fly four people, including two members of the Italian Air Force and a researcher, to space in late September or early next month.

Virgin Galactic described that planned mission as its first focused on commercial research, with crew members studying how the transition from gravity to microgravity affects the human body, and conducting other tests. The company has said it would charge the equivalent of $600,000 per seat for research and astronaut training missions, and at least $450,000 per seat for space tourism flights.

The FAA’s role in space includes regulating launches and re-entries of space vehicles. During missions, the agency restricts airspace, seeking to avoid potential accidents. The FAA said in July it had activated a new system to track in real time launch and re-entry vehicles, saying the tool would help minimize the impact of airspace closures due to space trips.

The agency’s approach to space at times has drawn criticism. A flight planned by Space Exploration Technologies Corp. for late June was delayed one day after an aircraft entered into restricted airspace, according to a tweet at the time by Elon Musk, founder of the company also called SpaceX. He called that restricted area “unreasonably gigantic.”

Write to Micah Maidenberg at micah.maidenberg@wsj.com