Air pumps are mostly aesthetic. The introduce nearly no dissolved oxygen to the water. However, they may agitate it a bit and that might help.
I had air pumps in my tanks and shut them off as they caused water to splash upon and stain the glass covers of my tanks.
No, fish cannot survive with only air pumps. I see in your signature that you have a 44 gallon tank. Tell us what kind of filtration, etc., you are running in it.