Speaking to reporters at a crowded Austrian Grand Prix news conference, the two men addressed a "road rage" controversy that has dominated the headlines since the June 25 race in Azerbaijan.
"I still have the utmost respect for him as a driver and will continue to race him hard through the rest of the season," declared Hamilton, who said at the time that Vettel had "disgraced himself" by driving into him in Baku.
He said Vettel called him on the Monday after the race and then texted an apology, which he accepted.
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The championship leader had driven into the back of Hamilton's Mercedes while both were following the safety car in first and second places, waiting for it to return to the pits.
Vettel then pulled alongside, gesticulating angrily, before banging wheels.
The German, a four times world champion, later accused Hamilton of "brake-testing" him by slowing suddenly.
The car's telemetry subsequently showed that was not the case and Hamilton said the accusation was one he particularly wanted correcting.
Vettel was handed a 10-second stop-and-go penalty, and finished fourth. He was then summoned to a hearing in Paris last Monday where the governing body declared the matter closed.
Initially reluctant to dwell on the matter, saying he did not want to "pump this up more than it is already", the Ferrari driver on Thursday repeated the written apology.
"It was the wrong move to drive alongside him and hit his tyres," he said.
"I don't think there was any bad intention (by Hamilton). I don't think he actually brake-tested me. I was upset and over-reacted. I am not proud of the moment."
Hamilton, 14 points behind Vettel after eight races, missed out on victory in Baku after a headrest worked loose and he had to pit. He finished fifth.
Before Baku, the two world champions had made much of their mutual respect and the budding "bromance" seemed to be back on.
"It's nice to hear that we are able to move forward," Vettel said. "I think the respect we have for each other on and off track helps us in this regard."
(Editing by Louise Ireland)
