SpaceX’s launch of its mammoth Starship rocket on Thursday accomplished a set of ambitious goals that Elon Musk, the company’s chief executive, had set out before the test flight, the fourth.

Lifting off from SpaceX’s launchpad at 7:50 a.m. in South Texas, near Brownsville, Starship rumbled into the sky.

After it dropped away from the upper stage, the booster was able to gently set down in the Gulf of Mexico while the second-stage spacecraft traveled halfway around the world, survived the searing temperatures of re-entering the atmosphere and also made a controlled splashdown, in the Indian Ocean.

The flight was not flawless, and tough technical hurdles remain. The successes, surpassing what was accomplished during the previous test flight in March, offered optimism that Mr. Musk can pull off his vision of a rocket that is the biggest and most powerful ever and yet entirely reusable.

The outcome also helps validate the company’s break-it-then-fix-it approach to engineering, with steady progress since the first test launch in April last year when the rocket had to be deliberately destroyed when it flew off course.

“They are showing a capability to make progress more rapidly than we may have thought they’d been able to make,” said Daniel L. Dumbacher, executive director of the American Institute of Aeronautics and Astronautics, a professional society for engineers. “They’ve got a team that knows what they’re doing, has the capability is willing to learn, and just as importantly, is not beholden to past assumptions.”

If Starship can fly again and again, more like a jetliner than a conventional rocket, it could transform a global space launch industry that SpaceX already dominates.

Today’s flight is also likely encouraging for officials at NASA. They are counting on SpaceX to provide a version of Starship to take astronauts to the surface of the moon during NASA’s Artemis III mission, currently scheduled for late 2026.

Bill Nelson, the administrator of NASA, offered his congratulations on X, the social media site that Mr. Musk owns.

“We are another step closer to returning humanity to the Moon through #Artemis — then looking onward to Mars,” he wrote.

After reaching a peak altitude of about 130 miles, the Starship upper-stage vehicle fell back to Earth, as planned, and re-entered the atmosphere. Cameras on the spacecraft captured an vibrant glow of gases heating up beneath it.

At an altitude of about 30 miles, pieces started peeling away from one of the steering flaps near the top of the spacecraft, with the flap continued to work. The camera’s view then became obstructed when debris cracked the lens.

“The question is how much of the ship is left,” said Kate Tice, one of the hosts of the SpaceX broadcast.

Real-time data continued to stream back, relayed via SpaceX’s Starlink internet satellites, to the company’s headquarters in Hawthorne, Calif., all the way until the altitude was reported at 0 — the surface of the Indian Ocean.

A final engine burn flipped Starship to a vertical position just before landing.

“From South Texas to the other side of the Earth, Starship is in the water,” said Dan Huot, one of the other SpaceX webcast hosts. “What a day.”

A crowd of onlooking SpaceX employees outside mission control in California cheered wildly, with arms thrust upward in celebration.

“Despite loss of many tiles and a damaged flap, Starship made it all the way to a soft landing in the ocean!” Mr. Musk wrote on X.

The damaged flap and the loss of heat-resistant tiles points to crucial upgrades still needed. Otherwise, Starship would, like the space shuttles, require extensive refurbishment after each flight.

“But that’s all fixable,” Mr. Dumbacher said. “It’s a step in the right direction, and there are more steps that have to be taken.”

Earlier in the flight, the rocket’s first stage, the giant Super Heavy booster was also able to perform maneuvers that in the future would take it back to the launch site. For this flight, it simulated such a landing by setting down in the Gulf of Mexico. All three previous attempts at that feat have ended in explosions.

With the Starship vehicle stacked on top of the Super Heavy booster, the rocket is the tallest ever built — 397 feet tall, or about 90 feet taller than the Statue of Liberty, including the pedestal.

The Super Heavy has 33 of SpaceX’s powerful Raptor engines sticking out of its bottom.

As those engines lift Starship off the launchpad, they generate up to 16 million pounds of thrust at full throttle. On this flight, one of the engines failed to ignite, but that did not prevent it from continuing its trip to space.

A couple of weeks ago, after a successful launch rehearsal, Mr. Musk wrote on X that for this flight, “Primary goal is getting through max re-entry heating.”

In other words, he did not want the vehicle to burn up. And on Thursday, it did not.

The Starship launches have attracted spectators to SpaceX’s launch site near the southern tip of Texas.

On Thursday, they sat in beach chairs or atop pickup trucks and camper listening to the SpaceX broadcast. as the countdown continued.

“It is insane what they are doing here,” said Chris Thomassen, who had traveled from the Netherlands to watch the launch, camping out three days on a beach close to the launchpad, then moving to a spot just at the edge of the safety exclusion zone.

Robert Opel, 56, set up a tent outside the launch site four days before Thursday’s launch. He was so determined to see the liftoff from up close he had arranged to travel across the Rio Grande to Mexico, which is just a few miles from the launchpad.

“It is like all of your birthdays wrapped up into one,” Mr. Opel said, adding that this was the fourth — of four — Starship test launches that he had witnessed.

Eric Lipton contributed reporting from Boca Chica, Texas.

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