1- in your sentence "He's immersed in his work"
the word immerse ( here ) means = to completely involve in work
Other Examples :
She immersed herself in her work.
Clare and Phil were immersed in conversation in the corner.
....................
2- immerse has another meaning which is :To cover completely in a liquid; submerge.
and
immerge : To submerge or disappear in or as if in a liquid
* * you asked if the people use "IMMERGE" to express the meaning of the word " IMMERSE " in the second meaning ( not in your sentence )
will immerge and immerse are Synonyms
and immege and immerse express the meaning of submerge ( cover in water ( liquid ))