Havertz's 64th-minute penalty was enough to earn Leverkusen a 1-0 victory in the match on Saturday.
However, the goalscorer was on the receiving end of Ascacibar's saliva as the midfielder spat on him in the 91st minute.
The Argentine was sent off for the incident, and he has now been given a ban for his actions which will extend until May 25.
"I have apologised for my behaviour with the club and the team," Ascacibar said on Tuesday.
"Of course, my apology is also true for my opponent Kai Havertz, even if I was provoked before, that should not happen from me."
Ascacibar was also widely criticised for his behaviour after the match, including from his own manager Markus Weinzierl.
"Of course, that's not possible," he said. "He has done the team a disservice, he must have his emotions under control."
Leverkusen manager Peter Bosz was equally unimpressed, stating: "In my opinion, that's the worst thing you can do in football, you can certainly play hard - but you do not do that."
This six-week ban means the 22-year-old will now miss the remaining five fixtures of Stuttgart's Bundesliga campaign.
However, if the club fall into the relegation play-off battle, he will be able to play in the second leg of that encounter.
They are currently 16th in the Bundesliga on 21 points and, as things stand, they will be involved in the playoff.
However, the two sides meet on the final day of the season on May 18 in a potential season-deciding fixture.
Ascacibar's absence is likely to be a big loss in such a crucial part of the season.
He's played the full 90 minutes in 27 Bundesliga games so far this season, and he has also played three times for the Argentina national team since making his debut in September 2018.
He originally joined Stuttgart from Estudiantes back in 2017 where he first rose to prominence.