President-elect Donald Trump struck a conciliatory tone at a much-anticipated meeting with some of his most prominent opponents Wednesday, offering to work with Silicon Valley executives to foster innovation and support fairer trade deals.

Part public spectacle, part private discussion, the 90-minute meeting in Trump Tower in New York marked the first time Mr. Trump met leaders of some of the nation’s most valuable companies, most of whom had supported Hillary Clinton during the campaign, and several of whom were the targets...