The striker turned United down as he "understood that he is at the centre" of Bologna's project.
Arnautovic was also hailed as a "true champion" by Di Vaio.
In an interview with Corriere dello Sport, Di Vaio said: "No, no.
"Manchester United did not back down because of those fans, but only because Arnautovic has understood that he is at the centre of our project now and in the coming years.
"Marko behaved like a true champion, realising what he represents for Bologna as a team and a city."