On Friday, Virgin said the Federal Aviation Administration granted approval of Virgin’s full commercial space-launch license, opening the door for outside passengers to join flights. The approval followed a test flight on May 22.

Virgin Galactic shares soared about 30% on Friday, to $52.37 in midafternoon trading.

Blue Origin has applied for but hasn’t yet received a similar approval from the FAA, an FAA spokesman said. “As for all license application reviews, the FAA will make a decision when and if all regulatory requirements are met,” the agency spokesman said of Blue Origin’s application. A Blue Origin spokeswoman didn’t immediately comment.

As the nascent space-tourism industry comes closer to selling rides to the public, executives and regulators have begun new discussions about how such flights should be overseen. Since 2004, Congress has restricted the FAA from regulating the safety of commercial space flights, hoping to insulate the sector from compliance costs while it develops. That policy has been extended several times and now runs until 2023.

Under the current framework, the FAA is charged only with keeping the general public and other aircraft safe during commercial launches, as spacecraft blast off and traverse the airspace that airliners and helicopters use. Virgin Galactic qualified for the commercial launch license by demonstrating that its technology worked safely in test flights, the FAA spokesman said.

But the agency still has no role in certifying the safety of flights themselves in the way that it oversees airplane flights. Instead, passengers must sign waivers acknowledging the risks of such flights.

Like Mr. Bezos at Blue Origin, Mr. Branson has pledged to participate in a coming Virgin Galactic flight. The British billionaire founded Virgin Galactic in 2004 after amassing a fortune in the music and airline industries. He also has taken on adventurous challenges such as crossing the Pacific Ocean in a balloon and racing across the English Channel in an amphibious car.

Two years ago, Virgin Galactic went public in a deal with a special-purpose acquisition company, a way of becoming listed on a stock exchange that has since rapidly gained popularity.

Virgin plans to send passengers to space in a rocket-propelled vessel that is launched from a highflying airplane. After being dropped from about 50,000 feet in the skies over New Mexico, the spacecraft shoots above the Karman Line—the imaginary lower boundary of space—where passengers experience weightlessness for a few minutes before the spacecraft descends and lands on a runway.

The roughly 600 people who have signed up as future passengers with Virgin Galactic would likely pay less than $500,000 for a ride, Virgin executives have said.

Virgin hopes eventually to offer space flights from multiple locations. The company has also cited plans to take tourists to orbiting hotels and to use its vehicles for superfast long-distance travel.

The Virgin spacecraft is flown by a pair of pilots. Blue Origin’s capsule is fully automated and is launched vertically from the ground in Texas.

Virgin hasn’t set a date for Mr. Branson’s flight, which would follow additional testing of the spacecraft. Blue Origin plans to send Mr. Bezos to suborbital space on July 20, along with his brother, Mark Bezos, and the yet-unidentified winner of a charity auction who is paying nearly $30 million for the ride.

Because of the cost of space launches and the limited federal regulation, the two companies’ vehicles have only received a fraction of the testing that commercial aircraft go through before they carry passengers. The companies have said they are following rigorous safety protocols as they prepare to take passengers to space.

Virgin Galactic said its review of the May 22 test showed that the flight was a success. The test was the company’s third spaceflight with a crew and first-ever spaceflight from its Spaceport America site in New Mexico.

The company said the FAA approval and its test-flight review “give us confidence as we proceed toward our first fully crewed test flight this summer.”

Write to Matt Grossman at matt.grossman@wsj.com