Red Bull's Daniel Ricciardo won in Baku after a chaotic race which saw Ferrari's Vettel given a 10-second stop-go penalty when he steered into the back of race leader Hamilton's Mercedes under safety car conditions after a red-flag stoppage on lap 22.
The four-time Formula One champion seemed infuriated with Hamilton after the British driver slowed on the exit of Turn 15. After the initial collision, the German gesticulated angrily with both hands as he then pulled alongside his rival and hit the side of Hamilton's car.
After his penalty, imposed by the stewards for dangerous driving, Vettel recovered to finish in fourth, one place ahead of his British rival.
The episode left Hamilton incensed. "Deliberately driving into another driver and getting away pretty much scot-free as he still came fourth, I think that's a disgrace. I think he disgraced himself today," the Briton told Channel 4.
"Imagine all the young kids that are watching Formula 1 today and see that kind of behaviour from a four-time world champion. I think that says it all.
"If he wants to prove that he's a man, I think we should do it out of the car face to face.
"Driving dangerously, which in any way can put another driver at risk... we were going slow, if we had been going fast it could've been a lot worse.
"It definitely sets a precedent within F1 and it also does for all the young kids that are watching us drive and conduct ourselves.
"They've seen today how a four-time champion behaves and I think, hopefully, that doesn't ripple into the younger categories."
Hamilton also denied the accusation that he had deliberately "brake-tested" his rival, arguing that he was merely "controlling the pace."
"Like all the other restarts, I slowed down in the same spot," he said. "He was obviously sleeping and drove into the back of me.
"That wasn't, for me, an issue."
Hamilton began the race in pole and avoided the melee in lap one to establish himself as race leader, but his own progress was stymied when he was forced to cede his lead after pitting due to a loose head rest.
While Vettel's sanction saw him drop from second to ninth, he eventually recovered to extend his championship lead over Hamilton to 14 points.
Hamilton's disappointing race in Azerbaijan extends his underwhelming record in Baku, with the 32-year-old emulating last year's fifth-placed finish.
