Starbucks Corp. on Wednesday asked gun owners to stop bringing their weapons into its cafés, thrusting the coffee giant deeper into a national debate over firearms that has proven vexing for retailers and other businesses.

The request, made in an open letter from Chief Executive Howard Schultz, was a response to intensifying pressure on the company for allowing customers to openly carry weapons in states that permit it. Gun control advocates had criticized the company for its laissez-faire stance, while gun rights advocates...