What to do if my iPad doesn’t update software?and I Don’t want to use itunes to install an updates. Eventhough I still have 38.3 gig in the memory

What happens when you try and install an update? The iPad 4 will only update to iOS 10.3.3. See instructions here, Update your iPhone, iPad, or iPod touch - Apple Support and troubleshooting here, Get help with over-the-air iOS updates - Apple Support

Are you sure that is an iPad 4? The iPad 2 & 3 will only update to iOS 9.3.5.

Yes.

Are you sure you own an iPad 4th generation?

Easiest way to tell if you have an iPad 4th gen or not (if it wasn't one of the very early models ) is if the charging connector port is the smaller lightning connector type OR if has the older, much larger 30 pin charging connector, that is an iPad 2 or 3 model.

The iPad 2 and 3 models cannot update past iOS 9.3.5.