You could,... but I would say "...in a friendly way". This sounds more natural. The word "fashion", in most contexts would sound a little old-fashioned, literary and/ or pretentious.

Yes, "He smiled at me in a friendly fashion" is a grammatically correct and clear sentence. It effectively conveys the idea that someone smiled at you in a friendly manner. If you're looking to vary your word choice, you could also consider using synonyms, like: He gave me a friendly smile. He grinned at me in a friendly way. He greeted me with a warm smile.