The European Parliament approved a major part of the trade deal that it settled on with the United States last summer at President Trump’s golf course in Turnberry, Scotland — nearly a year after the agreement was struck.

European Union lawmakers agreed to cut tariffs to zero on American industrial goods, including machinery and car parts, and on lobster. It will also reduce levies on some American agricultural products. In exchange, tariffs on many European goods exported to the United States should remain taxed around 15 percent (but sometimes more).

The decision ran up against the clock. Mr. Trump, frustrated with Europe’s slow progress toward agreeing to the deal, had threatened “much higher” tariffs on European products if it wasn’t finalized by July 4.

“With this milestone, we are days away from fulfilling our commitment to remove tariffs on imports of US industrial goods,” Ursula von der Leyen, the president of the European Commission, posted on social media after the vote.

European Union member states still need to rubber-stamp the agreement before it is officially complete. But Tuesday’s vote should effectively finish what has been a long and sometimes fraught process.

Earlier votes on the deal were postponed, first over Mr. Trump’s threats to annex Greenland, and later after the Supreme Court ruled that many of Mr. Trump’s tariffs were illegal.

Even once progress resumed, bumps continued. In an interview with The New York Post published on Monday, Mr. Trump threatened 100 percent tariffs on French wine and Champagne over digital taxes.

Parts of the deal can be suspended if the United States does not hold up its end of the bargain, as a way to give the European Union both a sort of escape hatch and some remaining leverage. Other parts will sunset after 2029 if they are not renewed.

The news came as European Union and national leaders gathered in Évian-les-Bains, France, alongside Mr. Trump for a Group of 7 meeting.