“Now we have very clear rules,” Mr. Breton said. “They can do whatever they want outside of Europe…but in Europe they will just have to fulfill [our] rules and obligations.”

The Digital Services Act aims to force platforms to take more responsibility for the content their users post online. It will introduce a series of obligations for large social-media networks like Twitter, ranging from new standards for taking down illegal content to a requirement to conduct annual risk assessments detailing how they are responding to harmful content that is posted online.

The text of the new law still needs formal approval from the EU Parliament and representatives from EU countries, but it is unlikely to change substantially from what lawmakers agreed to last week.

Mr. Breton expressed confidence in a post on Twitter earlier Tuesday that the company under Mr. Musk’s ownership would conform to the EU’s new rules.

“Be it cars or social media, any company operating in Europe needs to comply with our rules—regardless of their shareholding,” Mr. Breton wrote. “Mr Musk knows this well. He is familiar with European rules on automotive, and will quickly adapt to the Digital Services Act.”

Mr. Musk’s purchase of Twitter could mark an eventual test case for the EU’s new legislation once it comes into effect, said Carl Tobias, a law professor at the University of Richmond. While there are questions about whether the European Commission will have the resources to fully enforce the new law, Prof. Tobias said, “I do think [Mr. Musk] is on notice, and Twitter shareholders, too.”

Mr. Musk wrote on Twitter Tuesday afternoon that his view of free speech is speech that “matches the law.”

“I am against censorship that goes far beyond the law,” he said. “If people want less free speech, they will ask government to pass laws to that effect.”

Mr. Breton said the EU was careful in setting its regulations to maintain freedom of speech. He said he had interactions with many platforms during the lawmaking process and was open to meeting with others, in a comment that appeared to be aimed at Mr. Musk. “If any new shareholder of these platforms is willing to engage with me, my door will be totally open, like always,” he said.

The EU’s new legislation won’t prevent Mr. Musk from taking a more free-speech-oriented approach, said Zach Meyers, a senior research fellow with the Centre for European Reform think tank. However, he said Twitter will still need to comply with rules on the removal of illegal content, which would likely require hiring more people to deal with content moderation and complaints.

Write to Kim Mackrael at kim.mackrael@wsj.com