Hand sanitizer in baby's eye?

After changing my 15w old baby's diaper, I sanitized my hands and for some reason when I clicked the cap back on, a drop flung into his eye. Like I'm talking about one of those stupid, one shot in a lifetime things that somehow hit the most impossible targets.

He screamed so I quickly ran him to the bathroom and poured cold water from a wash cloth right into his eye a few times. He quickly calmed down and the redness quickly disappeared but I can't help but feel like absolute shit for this. I held cuddled him and gave him extra love till he fell asleep.

I Googled everything I could and all the results said the same thing, "just rinse it out with cold water and it will be fine". I'm still worried whether or not this would cause any damage in his little baby eye, even if it's not showing anything?