I want to say that I accidentally got paint in my eye in a more formal sounding manner with a single word. Is there a word I can use to express this in a sentence like "Paint ....ed into my eye." to mean "it accidentally entered into/touched"?
Native speakers typically express the idea like this:
I got paint in my eye.
Paint got in my eye.
See here, for example.
More formal and more generic ways of expressing the idea don't usually involve a more formal verb:
... foreign body in the eye ... intraocular foreign body ... chemical splash in eye ... intraocular chemical burns ...
but you could say
Paint came in contact with my eye.
Paint contacted the ocular surface.
The latter, with "ocular surface", is something you might find in a medical report. It is so far removed from everyday idiom that it could have a comical effect if the context in which it is used was otherwise informal and conversational.
Similar to the suggested spatter, there is also the word splatter.
Splash with a liquid, typically a thick or viscous one.
Splash (a liquid) over a surface or object.
I usually consider paint a viscous liquid.
A spot or trail of a thick or viscous liquid splashed over a surface or object.
So, your sample sentence could be "Paint splattered into my eye."
I was going to say "no, there is no single word in common use to indicate accidentally getting liquid into eye", but really, got is the word.
Saying "I got paint in my eye" is acceptable (and very common) in all but highly stylized contexts where precision is emphasized, such as a legal affidavit or medical record. It would be assumed to be by accident because you said "I got…" rather than "I put…", but you could certainly say "I accidentally got paint in my eye."
You may be concerned that "I got" is usually considered very informal, but in this specific instance it is considered a very proper way to indicate that this happened without you intending it to happen while still maintaining an active voice.
The paint intended for a surface that doesn't make it there is commonly called "spatter".
As in: "I was painting the ceiling, and got spatter in my eye." ('On the floor','on my shoes'. etc.)
Most dictionaries will show "spatter" as a noun and a verb: "I thought it would be neater to use a paint roller - but, it spattered the paint everywhere.
Spatter is also often used for a spray of blood. AmE