'Can you tweet something like: "Unbelievable support yesterday and great effort by the lads! Hard result to take! But we go again!",' Anichebe wrote on Sunday afternoon.
The message was hastily deleted but not quickly enough, with social media users pouring their scorn on him.
Anichebe joined Sunderland as a free agent in September and has made three substitute appearances for the Black Cats to date.
David Moyes' side remain rock-bottom of the Premier League - without a win in their nine matches so far.
'We started the game sloppily but recovered very well,' captain John O'Shea told the club website after losing to West Ham. 'We started the second half well and could have been in front.
'We needed to see out the game but unfortunately we switched off a little bit.
'Now we have to pick ourselves up off the floor. We want to start winning games as quickly as possible.'