Payet is public enemy number one with the Hammers faithful after refusing to play for the club as he engineered a £25m move back to Marseille.
"Does he owe an apology? No. I wouldn't say that," said Bilic
"Players are always moving and I'm not going that deep into what he should say or shouldn't. He is gone, he went home, that was his wish and I will not forget how good he was for us.
"He was brilliant. I was so happy and proud, with my team, my staff, to help him achieve what he did. What he achieved with us was a great story for West Ham, for the Premier League and for France.
"I ain't going to forget it just because he moved. I really wish him well."