Suarez apologised first via his Twitter account and then through a statement issued on the club's website for the incident, which happened before he scored a dramatic equaliser seven minutes into stoppage time to make the final score 2-2 at Anfield.
The club also issued an "unreserved apology", saying the 26-year-old's behaviour was "not befitting of any player wearing a Liverpool shirt".
There have since been calls from some quarters for the Anfield club to sack the star striker, who was last week shortlisted for the Professional Footballers' Association Player of the Year award, but Barnes believes that would be a step too far.
The former England international told BBC Radio Five: "He has to suffer the consequences of his actions - that's all he has to do.
"I've heard talk about him not playing for Liverpool again, which I think is ridiculous. I've heard talk about him bringing the good name of Liverpool down, which is also ridiculous - he's brought his own good name down.
"If you're saying because of what he's done he should never play for Liverpool again, who should he play for?
"Whoever signs him, it's going to reflect badly that they sign a player who Liverpool deem from a moral perspective not good enough.
"So Manchester United, Real Madrid, Barcelona... they can sign him? But we can't have him? If you're questioning his moral character, what you're actually saying is that if he can't play for Liverpool he can't play for anyone, which is ridiculous.
"The reflection on Liverpool is their response to the actions and their response is right. The response from (managing director) Ian Ayre and the club is 'it's completely unacceptable and we will not accept it'."
- Sports & Recreation