Did Nick really push Amy?

Amy's version:
1) She wanted a baby, Nick didn't
2) Nick is very harsh to her and pushes her down and leaves.

Nick's version:
1) He wanted a baby, Amy didn't
2) (This is the question that needs to be answered)

Let's bounce to the end of the movie when Nick gets angry enough to back Amy up to the door, HARD. This indicates that he is capable of hitting Amy. Sure, he was pushed to the limit and he reacts that way. But he is quite capable of hitting Amy.

Everything in Amy's journal is not a lie. The initial portions are real and slowly moves to her falsified version. This part where Nick pushes Amy down is somewhere in between. The argument Amy talks about in her journal is - how she wanted the baby but Nick didn't. The actual argument, if an argument did happen, would have happened the other way around - where Nick wanted a baby but Amy didn't. However, by this time Nick was banging another chick. Their arguments have been discussed to be not as intense, not enough for him to physically attack Amy over the want of a baby. It is shown that Nick has become distant and doesn't really care for Amy much. Besides, Amy needed a strong reason to backup the purchase a gun. She comes up with Nick's violent act to declare that she was scared of Nick, scared enough to buy a gun.

These elements point in the direction that Nick did not push Amy down though he was capable, he never cared enough to argue that intensely.

Here's a link that talks about other big plot-holes in the movie: Gone Girl's Plot Holes Explained

I would wager a no, he didn't really push her answer. This could be extrapolated out to also be no, he didn't push her as hard/in the way she denotes in the book.

Amy was writing the journal, all along, in an effort to get away with framing Nick for her murder, and as such anything she's narrating until we learn that it's all part of an elaborate ruse she's concocted should be seen as completely questionable/pure lies.

To that end, Nick may have pushed her, he may not have, but that's not really the point of the story. The movie isn't very interested in what actually happened, just the narrative of events that supposedly happened. Gone Girl is very much a critique on modern society's obsession with stories such as the one depicted in Gone Girl, and it's more about, "Fitting the narrative the media wants to portray," more than being a factual retelling of actual events. The media want their story, something juicy to keep people's eyeballs on their channel and website and other media offerings. They're willing to provide whatever story necessary to get their pound of flesh. This can be further expounded on by remembering that the movie only shows events in the past with narrative flashbacks, delivered as though they were being read as part of a narrative.