Learn Turkish: Onları gizlice gözetlemeden duramadım. - I couldn't stop myself from spying on them secretly.

You are exactly right! -mEdEn does mean "without". The entire phrase -mEdEn duramamak literally translates to "to be unable to stand/stay without doing [something]." In Turkish, this is the standard idiomatic way to say "I couldn't help but do [something]" or "I couldn't stop myself from doing [something]."

What is the exact grammatical difference between durmadım and duramadım?

The single letter -a- changes the entire meaning. Dur-ma-dım is simple negation meaning "I didn't stop." Dur-a-ma-dım includes the inability suffix (-e/-a inserted before the negative suffix), which changes the meaning to "I couldn't stop."

Could I just say gizli gözetlemeden instead of gizlice?

Yes, gizli can function as both an adjective ("secret") and an adverb ("secretly"). However, adding the -ce suffix strongly emphasizes the adverbial manner ("in a secretive way"). When describing sneaky actions like spying, gizlice sounds much more natural and expressive than just gizli.

In English we say "spy on them". Why is it onları (accusative) instead of taking a dative or locative case?