FIFA has opened disciplinary proceedings against the striker after he appeared to bite the shoulder of Giorgio Chiellini while playing for Uruguay in their 1-0 victory over Italy at the World Cup on Tuesday.
If found guilty, this will be the third time that Suarez has bitten an opponent following incidents at Ajax and then Liverpool, when he sank his teeth into the arm of Chelsea defender Branislav Ivanovic last year.
Fowler believes that Liverpool may run out of patience with the 27-year-old after they backed him following the biting incident, and after he was found guilty of racially abusing Manchester United defender Patrice Evra.
"Most Liverpool fans love him as a player but he's continually dragged the club's name through the mud," Fowler told talkSPORT. "It's not right, especially after how they helped him last time. They tried to rehabilitate him. I wouldn't be surprised if Suarez left now.
"He's a different person on the pitch. I'm flummoxed, he's dragged the good name of Liverpool through the mud again, and it's not right. You can't defend him."
Before the incident, Suarez, who could reportedly face a two-year worldwide ban, was linked with a summer move to Barcelona.