There's no "war on Christmas"-level controversy surrounding the greeting (it means "Happy Ramadan"). Your Muslim co-worker will appreciate the thoughtfulness.
Ramadan is now at the doorstep or getting closer to us. In various civilization and cultures, Muslim communities have a great tradition to wish Happy Ramadan and other related duas. Wishing Ramadan to friends, families, communities, and neighbors is also a Sunnah and practiced by Prophet Muhammad (peace be upon him).
During Ramadan, you may notice that Muslims greet each other and utter the Arabic phrase “Ramadan Kareem”. You may be wondering what Ramadan Kareem means. So, what is Ramadan Kareem? It is basically used as greetings among Muslims. This is used to wish their fellow Muslims a happy Ramadan. You can also utter Ramadan Kareem when you want to wish your friend to have a generous Ramadan.
Ramadan is spiritually and religiously precious event in the Islamic history and thus for Muslims. Ramadan basically connects creation to its creator in the most beautiful way which spiritual value is countless. That is why Muslims blissfully and faithfully greet others/ family/ friends saying Ramadan Kareem or Ramazan Mubarak or Happy Ramadan.