T., you are correct - children can develop fears like this for no reason. My daughter started freaking out like I was throwing her to the wolves when I put her in the bathtub, suddenly, no warning, just a couple months older than your son. My (now ex) husband started coming in and badgering me about what I was doing to her to make her scream. Finally he came in and watched from bath prep to getting into the bath and saw there wasn't anything traumatic going on, just her response. It's just one of those wacky developmental things that are not uncommon. I chalked it up at the time to maybe a bad dream she had about a bath, but it's so widespread among kids that age, who knows? New independence/assertiveness phase? Anyway, I kept at it, calmly putting her into the tub over and over, if she had to cling to me, fine, I'd wash her that way (learned to change into raggy-clothes before bathtime, no worries of her stretching or tearing or soaking my clothing). She got over it, but it did take a few months of persistence. Be glad you have a boy - hair washing was the trickiest part!
So, you're not a bad mommy, you are not scarring him for life (it's hard to get my now seven year old out of the bath these days, she loves to lounge and soak in the warm water), and whatever gets washed, gets washed, no harm done if he doesn't get every square inch sanitized every bath time. Your consistency and calmness is most important, so he'll see that it's not negotiable, and mama isn't worried, so he'll eventually pick up on your vibes that it's no big deal and you're going to wash him, regardless. So no trying to persuade him how safe it is (it's not an intellectual issue for a 22 month old), don't plead or cajole, just show by example that you don't see anything out of the ordinary and keep washing however you have to, even if he's clinging to you like a (very loud) tick. If you can sound bored, but descriptive of what you're doing ("I'm washing your back now, wash wash wash, rinse rinse rinse"), it may help calm him, too.
Good luck, and I hope you don't lose too much hair over this phase (either by you or your son pulling it out!).