You know what, you're right! In this situation, a graecophone would always say "Συγγνώμη" (which is the correct spelling, or, sometimes, as jaye16 notes, just "sorry"!) He would say "Λυπάμαι" in a situation where an anglophone would say "I'm sorry". I know that "I am sorry" means "Λυπάμαι", but I think "Sorry", on its own, can also mean "I apologise", isn't that right?
"συγγνώμη" means "excuse me or forgive me" You can use it for example if you interrupt someone etc. or get someone's attention. But also if you have done something and want to be forgiven or excused. "λυπάμαι" literally means 'I'm sorry' e.g. I feel bad (sorry) for a problem, a worry etc you have.
Ok, I'll leave you some links with info about GReek and theGreek course on Duolingo. But first a note: Greek pronunciation is really straight forward with each letter pronounced as it appears in the alphabet. Ok so there are a few excecptions.
So, the word "Συγγνώμη" has a clear S sound at the start. υ=εε "γγ=g" (that "g" is rather muted) then νω = "no" and μη ="me" (as in 'you and me).
Ok, check this for native speakers:
And these for general knowledge: FAQ: