The Gabon international finalised a transfer to the Emirates on deadline day last month in a deal that is rumoured to cost close to £60m.
In the weeks prior to the transfer, Aubameyang was suspended for disciplinary reasons and went on to miss two matches.
Watzke has now suggested that the forward purposefully did not put in any effort with the team in order to force through a move to the Premier League club.
"On Thursday, I laid it on the line for the squad - the next player who does something like that will fail miserably," the businessman told FAZ. "He'll run into major problems. The next player who tries to put us under pressure by withholding performances or even going on strike will not get a pass and will instead sit in the stands.
"They are all aware of it. This is now a public statement. Following Dembele and Aubameyang, there is now an overriding interest for the club - we need to get a better structure into our squad again. Aubameyang committed himself to us for this season, especially as a reaction to Dembele's transfer.
"But he also had the notion that we owed to him because in 2016 he opted for us and decided against very lucrative offers amid the exits of [Mats] Hummels, [Ilkay] Gundogan and [Henrikh] Mkhitaryan."
Aubameyang scored his first Arsenal goal in a 5-1 rout over Everton on Saturday.