Incident occurred in ny

If you were going against the flow of traffic, it was your fault.

I'd guess the backing-out driver is guilty of not making sure the way was completely clear. The place where it occurred has nothing to do with it; the same would be true in Illinois.

Does anyone have any ny case law on who would be liable: the car reversing in the traffic lane at a shopping center or the person reversing out of parking space?

The person reversing out of parking space, under law, has to be sure that the lane of traffic is clear of "on coming" traffic.
They would not even be looking at a car or for a car traveling the wrong way in the lane ( going in reverse)

Part of the issue not explained is why the car was going in reverse in the flow of traffic.

The car reversing would since they hold the problem of making sure then going the wrong way in the flow of traffic it is clear.

The court will look at specific traffic laws of the state.

The person going backwards was not literally in the street she was in a shopping center reversing so a car in front of her could back out of their parking spot

A Martindale-Hubbell lawyers site says this --

If you were backing out of a parking spot and hit a car, you would be at fault since you failed to use reasonable care when backing out of your spot. Even though the person may not have had their lights on or was driving very quickly, because they had the right of way, they're not at fault.

The car that backed up had the right-of-way, being in the traffic lane of the lot, and allegedly checked to make sure he could back up before he did, but didn't realize another car would back out of a parking spot and hit him (or be in his way).

What WG is saying is correct in most cases. But I'm not sure that would fly in this case. One point not mentioned, is whether the traffic lane was 2-way. If the traffic lane was 1-way, the driver backing out of the spot would have a reasonable expectation that traffic would only be coming from one direction.

Lets hear the full situation rather than bits and pieces. Were both cars insured? Did you turn this over to your insurance carrier? If so, what are they saying?