Does the letter of completion for degree (PhD) allow me to use the degree officially in Europe?
Perhaps you can apply for jobs already (explaining that you already finished your PhD). You could even receive a job offer, and start working on the immigration procedure.
However, many official applications (to the goverment, etc.) require you to submit a copy of your Diploma registered with Apostille. This is also true for many attractive immigration programs.
Don't forget to get this Apostille before departing from the US.