Hello Robert, and thank you for choosing JUSTANSWER.com
Its hard to know EXACTLY what the issue is.....but from what you are describing, it sounds like a hardware failure.
Most likely the Mobo......but it could be the RAM or CPU as well.
A failed HDD woudl say NO BOOT DEVICE, and a failed power supply......you wouldnt get anythign at all, most likely.
Did you have to remove any components when you replaced the battery?
The problem might be that you are using Windows 10, and your machine is locked up.
Your Gateway is NOT listed as a supported model that is eligible for a Windows 10 upgrade
See the image:
Only 2 models are listed and the DX4860 is NOT one of those.
Its quite possible a driver was envoked, and caused a freeze, or even a drive malfunction....its hard to say.
But, you shoudl see if you can BOOT from a different Media....like a DVD disc.
Using a different computer, make a BOOT CD, and see if the problem lies in the hard drive system files.
But honestly, this sounds more like a hardware failure than anythign else, and you need to have the componenets individually checked.
You are now facing the reality that is: taking this computer into a shop for diagnosis.