It is very interesting how the Proto-Slavic word mъrkỳ has a lot of descendants, and carrot comes from Middle French carotte, from Latin carōta, from Ancient Greek καρῶτον (karôton), and also Spanish zanahoria or Portuguese cenoura has a different etymology with Ancient Greek origin (see the *note below.)
From Bulgarian морков (morkov), Russian морко́вь (morkóvʹ)
morcov m (plural morcovi)
Etymology (Bulgarian морков)
From Proto-Slavic *mъrky.
From Proto-Indo-European *mrk-uH-. Cognate with Old High German morha, Old English moru, Lithuanian morka.
*Note: the Portuguese cenoura has many other cognates as:
Galician cenoira, Mirandese cenoura, Asturian cenahoria, Spanish zanahoria, Catalan safanòria and Basque azenario
From earlier çanoira, from Old Spanish çahanoria (Modern zanahoria), from Andalusian Arabic *[script needed] (safunnárya), from Arabic إِسْفَنَارِيَّة (ʾisfanāriyya), from Ancient Greek σταφυλίνη (staphulínē) ἀγρία (agría).
I can just say Wow!!! :)
Romania is bordered by Bulgaria, Serbia and Ukraine, they also used to border USSR, Czechoslovakia, Russia (?) and Poland and Old Church Slavonic was being used for religious and administrative purposes in the past on Romanian teritories, so there were many directions the Slavic influence could come from. And Slavic languages are considered to be fairly close to each other, as they splitted into different languages from Proto-Slavic fairly recently (I think as late as 1000 A.D. all Slavs could still understand each other) so it's often hard to know for sure which one of them was the direct source of given word to Romanian. In Polish "carrot" means" marchew", where "w" is read the same way as "v" in English or Romanian, so as you see, it's still fairly close to the Romanian word, although I don't know other Slavic languages, so I'm not sure if it's even closer in some of them.