The 59th-ranked Watson, lifted throughout the match by a raucous partisan crowd, played the match of her life but fell short of becoming the first British woman to beat a No. 1 since Sue Barker defeated Chris Evert in Boston in 1979.
"I've had some tough losses, but that was probably my toughest match I've played (on Centre Court)," Williams said. "Heather playing in front of her home crowd ... I think she played unbelievable. She should have won the match at this point, she was up two breaks and she just really gave her all."
Venus Williams, also a five-time champion, made it to the fourth round for the first time since 2011 by beating Aleksandra Krunic of Serbia 6-3, 6-2.
That sets up the 26th career meeting between the sisters - and the first at a Grand Slam since Serena won in the 2009 Wimbledon final. Overall, Serena leads 14-11.
"She's in better form than I am," Serena said of her older sister, "so I think she has a little bit of an advantage going into that match. But at least one of us will be in the quarterfinals, so that'll be good."