Verstappen is now 30 points behind Hamilton in the drivers' standings, making him the closest challenger to the 35-year-old as he bids to secure a record-equalling seventh title.
Mercedes have once again been the dominant force this season and were expected to race off into the distance after they locked out the front row at Silverstone.
But Hamilton and team-mate Valtteri Bottas finished second and third after hot temperatures both on the track and on the tyres worked against the German manufacturer.
Up until Sunday's race, Hamilton looked on course to home in on Michael Schumacher's record of 91 race wins, but Verstappen's drive ensured the pressure is still on Mercedes ahead of the Spanish Grand Prix next weekend.
"It's great that Max won. As I said before I want to have races and a challenge," said Hamilton.
"Obviously the Red Bull seems to be quite close to us in race conditions, but that's a good thing, and it will be interesting to see their progression through the rest of the year.
"I definitely will not overlook them. We will keep a close eye on them. It's not going to be easy by any means.
"I don't think my team has that mentality anyway. And maybe in the future we will get to have a bit more of a race."
Additional reporting from the Press Association.