T’would seem not – AH gives the Middle English couren, of Scandinavian origin as the origin for cower, whereas coward comes from the Latin for tail:

NOUN: One who shows ignoble fear in the face of danger or pain.
ETYMOLOGY: Middle English, from Old French couard, from coue, tail, from Latin cauda.
OTHER FORMS: cow ard —ADJECTIVE
WORD HISTORY: A coward is one who “turns tail.” The word comes from Old French couart, coart, “coward,” and is related to Italian codardo, “coward.” Couart is formed from coe, a northern French dialectal variant of cue, “tail” (from Latin c da), to which the derogatory suffix –ard was added. This suffix appears in bastard, laggard, and sluggard, to name a few. A coward may also be one with his tail between his legs. In heraldry a lion couard, “cowardly lion,” was depicted with his tail between his legs. So a coward may be one with his tail hidden between his legs or one who turns tail and runs like a rabbit, with his tail showing.

American Hurtage

That is, unless there's an unrevealed deep level of connection between the Scandinavian root and Latin, which seems a bit unlikely.