My eye is bloodshot and hurts when I touch outside of eyelid. No itching. Should I see doctor?

Yes. Whenever there is pain associated with redness in an eye it is advisable to see your eye doctor as soon as possible.
Episcleritis. Yes you need to see your ophthalmologist. Red eye which is painful to the touch could be episcleritis which is usually not a very serious problem. It could be other things also so check itout.