Found my iPhone 6 screen separated, how to repair it?
Yes, the swollen battery could have pushed the display outwards (the battery is just beneath the display)
I would replace the battery with this guide:
https://www.ifixit.com/Guide/iPhone+6+Battery+Replacement/29363
Be very careful when replacing lithium batteries as they are very flammable.
I've changed hundreds of them and still get caught out occasionally by those feisty things (due the strong adhesive)!