It is appropriate to say "Happy Ramadan". It is not exactly a holiday though, it is the ninth month in the Muslim calendar and as you may know this calendar is based on the moon so it is not fixed. Ramadan like all other months of this calendar can be observed anytime in the year, months are being shifted like 10 days each year compared to the 365 days calendar that is based on the Sun. During this month those who practice Islam and have the ability should fast from sunrise to sunset. The first day of the following month is being observed as the holiday of Ramadan, which is called "Fitre". On this day Muslims celebrate their ability to follow order of God to fast for a month and every Muslim family gives out food to poor. The amount of food depends on the number of family members. Also on this day nobody should fast. It is said that Quran has been presented as a whole book to the prophet in this month.

I'm not practicing Islam but I don't think that "happy Ramadan" really fits. It is exactly like "Day of Atonement" in the Jewish practice and it is more solemn than anything. I think it is more appropriate to say "happy Fitre" at the end of Ramadan, that fits better but Muslims would not being offended by saying "happy Ramadan" either, actually what lately I've seen them doing is to promote "Happy Ramadan" more than ever. That was just my personal opinion of someone who has been in direct touch with this religion.