Can a Catholic priest study nursing? What relevance is it to him and his cherished vocatioon?
That would be something for the priest to discuss with his bishop or religious superior.
Yes, priests can and do continue to earn degrees even after they are ordained. But that is going to be up to the one to whom the priest owes obedience. If there is a case to be made for how that education will benefit his service as a priest, then he can make that case and his bishop or superior will make the call.
Even apart from formal education, all priests do have ongoing formation in the form of retreats and other workshops that their bishop or superior requires them to participate in.
Is it a missionary order?
Why is this an issue for you?
Short answer is “yes, it’s possible”.
Longer answer, it’s unlikely, especially for diocesan priests. That being said, I do know a couple religious priests who went on to receive nursing degrees after ordination. Ultimately, it comes down to the Ordinary (Bishop or Religious Superior), and relevance to the priest’s ministry (E.G. Hospital chaplaincy, or providing services to the poor).