leech (n.1) Look up leech at Dictionary.com
"bloodsucking aquatic worm," from Old English læce (Kentish lyce), of unknown origin (with a cognate in Middle Dutch lake). Commonly regarded as a transferred use of leech (n.2), but according to OED the Old English forms suggest this is a distinct word, which has been assimilated to leech (n.2) by folk etymology. Figuratively applied to human parasites since 1784.
leech (v.) Look up leech at Dictionary.com
"to cure, heal," c. 1200, from Old English also had a verb læcnian, from the source of leech (n.2). Meaning "to apply leeches medicinally" is from 1802 (implied in leeching), from leech (n.1). Related: Leeched.
leech (n.2) Look up leech at Dictionary.com
"physician" (obsolete, poetical, or archaic), from Old English læce "leech," probably from Old Danish læke, from Proto-Germanic *lekjaz "enchanter, one who speaks magic words; healer, physician" (source also of Old Frisian letza, Old Saxon laki, Old Norse læknir, Old High German lahhi, Gothic lekeis "physician"), literally "one who counsels," perhaps connected with a root found in Celtic (compare Irish liaig "charmer, exorcist, physician") and Slavic (compare Serbo-Croatian lijekar, Polish lekarz), from PIE *lep-agi "conjurer," from root *leg- (1) "to collect, gather," with derivatives meaning "to speak (to 'pick out words')."

For sense development, compare Old Church Slavonic baliji "doctor," originally "conjurer," related to Serbo-Croatian bajati "enchant, conjure;" Old Church Slavonic vrači, Russian vrač "doctor," related to Serbo-Croatian vrač "sorcerer, fortune-teller." The form merged with leech (n.1) in Middle English, apparently by folk etymology. In early Middle English also of God and Christ; by 17c. the sense had so deteriorated leech typically was applied only to veterinary practitioners, and soon it was entirely archaic.

The fourth finger of the hand, in Old English, was læcfinger, translating Latin digitus medicus, Greek daktylus iatrikos, supposedly because a vein from that finger stretches straight to the heart.