Since the 1970's, the Earth has been warming. Can this be due to natural, increased solar output? Why or why not?

No.

The Sun can influence the Earth's climate, but it isn't responsible for the warming trend we've seen over the past few decades. We know subtle changes in the Earth's orbit around the Sun are responsible for the comings and goings of the ice ages. But the warming we've seen over the last few decades is too rapid to be linked to changes in Earth's orbit, and too large to be caused by solar activity.