East Stirlingshire's chairman expressed "deep regret" for the distress caused by the under-19 player's social networking posts.
The player, understood to be Brett Molloy, is now the subject of disciplinary procedures and police are also looking into the case.
The Ibrox disaster claimed 66 lives on January 2 1971 when fans were crushed on stairway 13 of the stadium at the end of an Old Firm game.
Rangers said it is "truly abhorrent that anyone could see fit to mock such a tragedy".
An apology issued by East Stirlingshire chairman Tony Ford has been published on his club's website.
It read: "I have been contacted by a number of people regarding posts made on social networking sites by one of our under-19 youth team players.
"Having read these comments I share your sense of shock and revulsion. Furthermore, I am horrified that a player connected with our club can express such opinions. I am sure all Shire supporters will join me in condemning these words.
"On behalf of everyone at East Stirlingshire Football Club I would like to apologise for the actions of this individual.
"As chairman of the club I would like to express deep regret for the distress these words will have caused the families of those who lost their lives in the Ibrox disaster. I would like to say sorry to all Rangers supporters for the grossly offensive nature of the descriptions used.
"And I would also like to apologise to the officials, staff, coaches and players of Rangers Football Club.
"The player concerned has already been spoken to by club staff and has been told he will be the subject of a disciplinary procedure."
The statement was reposted on the Rangers website.
A foreword to the statement from the Glasgow club stated: "East Stirlingshire have issued a full apology to Rangers Football Club following sickening posts made by one of their club's teenage players.
"Rangers condemns the sickening comments made and we note East Stirlingshire have issued a swift apology to everyone connected to the club."
Strathclyde Police said they are investigating. "We can confirm that a complaint has been received and inquiries are ongoing," a force spokeswoman said.