The three-time Champions League winner, who was released by Chelsea at the end of last season, has joined Everton on a two-year contract after being linked with several clubs.
The 33-year-old's ambivalent campaign at Stamford Bridge was best remembered for Mourinho's unguarded and unflattering comments about his age.
The veteran striker later made known his frustration about those remarks, labelling Mourinho a "fool" and a "puppet", but he now seems keen to consign the controversy to history.
Eto'o said on Wednesday: "The only thing I have to say vis-a-vis José Mourinho is to thank him.
"It is down to José Mourinho that I had the opportunity to be accepted and integrated into the Premier League, to play some great matches. That is the end of the discussion."
Of a possible debut against Chelsea at Saturday tea-time, Eto'o added: "I am not concerned about that aspect of it - that it is against my ex-team. The important thing is to integrate myself into my team-mates' way of working and get myself ready during the training and preparation, potentially, for that match.
"If that is not the case, should I not feature, it is important for me to get my head into the training for each match."