Aubameyang gave Arsenal a 12th minute lead but was sent off for an early bath in the 64th minute after Jordan Ayew had equalised for the home side.
Aubameyang was initially shown a yellow card by referee Paul Tierney for the tackle on the former Schalke 04 man but following a lengthy VAR review, the card was upgraded to red.
Meyer was unable to continue after the challenge as new signing Cenk Tosun took his place.
Arteta said: “I just watched it. It looked nasty to be fair. We know there is no intention to make such a tackle.
“Hopefully he (Meyer) hasn’t been injured and we are sorry for that. They made a decision and we have to accept it.
“It is a massive blow (to lose Aubameyang). You know how important Auba is for the team and we lose him for a couple of games.”
Aubameyang will now serve an automatic three-game ban and will miss upcoming games against Sheffield United, Chelsea and Bournemouth.
The Gabonese international, who is Arsenal’s top scorer with 14 goals, maintained he did not intend to hurt his opponent.
"Hopefully Max Meyer gets well soon," Aubameyang wrote on Instagram.
"I really didn't want to challenge him so hard and I appreciate that the boys fought hard to take that win but unfortunately we come back with a draw we keep going."
