Mullen's comments on the social media network in the early hours of the morning were quickly removed but not before they were retweeted.
The 19-year-old, who joined Hibs in the summer after being released by Hearts, wrote: "To the homosexual who ruined my night. Thanks you p**f bore off and grow a set."
In a statement on the club's official website, Mullen said: "I deeply regret what was a stupid and unacceptable tweet, and I want to apologise for any offence it may have caused.
"I realised my actions were wrong very quickly, and took down the tweet within minutes.
"The tweet has obviously been seen by some people and I accept that my choice of words was completely inappropriate."
A club spokesman added: "Fraser has apologised for his actions. The club cannot condone the content of his tweet, and will carry out its own investigation into all of the circumstances surrounding this and will take whatever action we deem appropriate."
The controversy came weeks after professional clubs in Scotland and England were sent rainbow laces by campaign group Stonewall in a bid to tackle homophobia in football.