Everyone has disagreements or arguements (fights). Some more than others. I personally have a pretty solid relationship with my wife of 12 years. We do fight from time to time. Sometimes it even involves some "choice" words.
What you don't want to do is hold grudges. I generally give her about 10 - 30 minutes to vent and get it out of her system. Once we've both calmed down, I'll give her a big huge and tell her I love you. Sometimes it's not that easy. As far as him thinking you are flirting....that's obviously jealousy on his part. There's nothing wrong with being jealous, just as long as it doesn't consume him. Jealousy can eat you alive. I know. I was an extremely jealous person when I was a teenager. As I got older, I grew out of it.
I'm sure he accused you of "flirting" because he felt threatened. Just re-assure him that you are "his" girlfriend and nobody can take you away from him. Let's just hope and pray he's not like I used to be. Very obsessive. That will push most girls away. Only time will tell just how "healthy" your relationship will become. At a young age, my relationships were not healthy, mainly because I was a very jealous person. Looking back now, it seems so silly that I was that way.