The 19-year-old last month posted a photo of his fellow Spaniard Enrique having his teeth whitened with the comment: "What f*** is he doing? This guy is gay... he does everything except play football."
Suso later deleted the tweet and uploaded the photo with a different comment, saying: "I dunno what to say..."
The FA on Tuesday responded by issuing Suso with the fine and a warning over his future conduct.
The teenager apologised on his Twitter page tonight, saying the "unfortunate" comments were intended as a joke.
"I'm very sorry if anyone has been offended by my comments about the photo of my team-mate Jose Enrique," he wrote.
"It was only a joke between the two of us. I realise that my words were unfortunate and it was not my intention to offend anyone.
"I apologise again if anyone was offended."
Enrique also took to the social networking site, describing his team-mate's punishment as "amazing" for what he had regarded as "banter".
"Is amazing how FA can fine my friend Suso Fernandez for a banter thing. Was just a joke!!!" he wrote.
In handing down the sanction, an FA statement read: "Following an independent regulatory commission hearing yesterday (Monday, December 17 2012), Liverpool's Jesus Fernandez Saez (Suso) has been fined £10,000 and warned as to his future conduct for a comment posted on Twitter.
"The charge, which the player admitted and requested a paper hearing, was that he acted in a way which was improper and/or brought the game into disrepute in that the comment was posted on his Twitter account and included a reference to a person/s sexual orientation and/or disability."
Suso has become an increasingly important member of the Reds' first-team and has made 13 appearances so far this season.
The attacking midfielder moved to Anfield from hometown club Cadiz in 2010 and signed a new long-term contract in October.