Verstappen was sent spinning off the track and into the barriers, with his car left in a crumpled heap.
The championship leader was able to walk away from the wreckage but was left "massively winded", according to team principal Horner.
It was the most sensational moment of the season, reflective of the intensity of the rivalry between reigning champion Hamilton and this year's leader in the standings.
Hamilton was looking to get ahead of his rival on the inside when wheels touched and Verstappen was sent spinning off the track, Horner denouncing what he considered "a desperate move".
Horner was furious with Hamilton, whose driving was scrutinised by the stewards as the race was red-flagged, with the cars returning to the pit lane while the barrier was repaired.
Verstappen was said to have come out largely unscathed, while Hamilton received a 10-second time penalty when the race eventually resumed.
"He's gone to the medical centre for a check-up. But that's a hell of a relief to see him get out because that corner's one of the fastest on the calendar," Horner told Channel 4.
"[It was] completely out of order [for Hamilton] to stick out a wheel on the inside there. It was way too far. Every driver that's driven this circuit knows you don't stick up the inside at Copse.
"He's done it and obviously his front left has made contact with Max's right rear, on one of the fastest corners in the championship and he's put him in the fence. Thank god he's not been hurt. As you can imagine, we're pretty annoyed about things.
"I think it was a desperate move. He failed to make the move in the first part of the lap which he was obviously geared to do.
"Then it was a desperate move sticking a wheel up the inside. Copse is one of the fastest corners in the world, you don't stick a wheel up the inside, that's just dirty driving.
"That's just not on and I'm just relieved to see our driver's walked away because that could have been a very, very nasty accident."