The 21-year-old Polish goalkeeper has been reminded by the club of his responsibilities, and apologised for any offence his remarks had caused.
The statements mocking Wales captain Ramsey have since been deleted from Szczesny's Twitter timeline.
"I'm sorry if anyone was offended by my tweets," Szczesny wrote on Twitter.
"There are some things we should not joke about and I have crossed the line. Sorry!"
Poland international Szczesny joined the Gunners in 2006 from Polish side Legia Warsaw and has since made 35 appearances for the club - including their 1-0 FA Cup victory over Leeds United on Tuesday.
Ramsey, also 21, had posted a picture of himself attending a golf event, to which Szczesny responded: "I don't wanna be rude mate but you look like a rapist on that picture."
The Wales international had treated the comments as light-hearted, replying: "I try my best mate we're not all blessed with a good fashion sense like you".
Thu Jan 21 2021, 20:01 by BazSpur
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