Is Julie and I proper grammar?

The noun phrase "me and Julie" is the correct objectiveform. The pronoun 'me' functions as the object of a verb or a preposition.

Examples:

Marvin invited me and Julie to lunch. (direct object of the verb 'invited')

Marvin sent me and Julie some flowers. (indirect object of the verb 'sent')

Marvin is the brother of me and Julie. (object of the preposition 'of')

Note: The term 'me and Julie' is correct grammar but it is traditional courtesy for the speaker (me) to put the pronoun last in a pair or group of people named (Marvin invited Julie and me...)