It depends upon the specific location and how far back behind the lines we're talking. As you noted the movie portrays the Omaha Beach landings. With the possible exception of Pointe du Hoc, Omaha saw the fiercest and most costly fighting of the five designated beaches. The other four beaches (Gold, Juno, Sword, and Utah) saw varying degrees of resistance, with Utah being the least tenaciously held.

It's important to note that the Allies also dropped paratroopers behind the beaches (as portrayed in the movies) in order to capture key crossroads and causeways, knock out enemy artillery positions, and in general cause mass amounts of confusion. There's quite a bit of debate about how effective these drops were. The general consensus of airborne operations throughout the war is that they could be effective and contribute to the success of an operation, but the costs were almost always staggering, often prohibitively so.

In regards to JUST the beach resistance, it was usually the case that once German fortifications had been flanked, the occupants would either retreat or surrender. As a rule of thumb, soldiers greatly prefer those two options to fighting to the death. However that's not to say it didn't happen. I've read at least one account of a German defender calling down artillery on his own bunker rather than surrendering or retreating.

Another key component to remember is Hitler's reluctance to commit additional units to the Beaches' defense. He believed that the landings were merely a diversion, and regardless preferred allowing the Allies to make their way inland a bit before counterattacking. What might have happened if he had committed additional units wanders into the territory of "what if?", so I'll not comment on those possibilities.

It's important to understand that the Germans had excellent defensive terrain behind the Beaches. Allied reconnaissance had failed to understand just how effective the Normandy hedgerows were as impromptu fortifications. They differed dramatically from British equivalents, inasmuch as the hedges were much taller and hardier than British hedges. The Allied soldiers that conducted the landings had been trained magnificently in how to breach the beaches, however their training was almost completely lacking in what to do AFTER those Beaches had been secured. It resulted in months of bloody and costly on-the-job training.

On a final note, any historian worth their salt is naturally suspicious of Hollywood films. They are entertainment, not documentaries (and even documentaries need to be viewed with a critical mind). SPR likely gives a vague approximation of the fighting and some of the difficulties encountered, but the only people who will ever truly know "what it was like" were the people that fought in the battle.

Russell Millers "Nothing Less Than Victory" consists of nothing but first person accounts of the landings and immediately afterwards. Some are only a paragraph long. It is most fascinating when he has accounts from different sources at the same place.