Not quite. The word "tedesco" comes from "theodisce", yes, but it itself is a borrowing from Germanic "þeod", meaning "people, race, nation".
By the way, Spanish "Alemania", French "Allemagne", etc. ultimately derive from the Germanic word "confederation", literally "All-Men".
All the other replies are very good- I just thought I would add this: In English, the commonly-used word is of Latin origin ('germanus'), but there also exists "teutonic" for certain contexts, and in the other Romance languages there are words similar to the Italian one (Spanish= 'tudesco'; French= 'tudesque') but they are not used.