Webster says that the word "slave" goes back to the word "Slav", as the early slaves in Europe were all from among Slavs. Is this etymology generally accepted, or are there some other theories?
This etymology seems fairly certain. Per the OED, the words Slav and slave comes from the Medieval Latin sclavus (c. 800CE), itself from the late Greek Σκλαβος (c. 580CE). According to the OED, documents of the ninth century attest to the attribution of word to the subjugated peoples of central Europe.
The origin appears to have developed from the wars of Otto the Great against the Slavs, ( people from central and east Europe) many of whom were reduced to the state of slavery as suggested by Earnest Klein. The above theory appears to be quite established, what remains unclear and has generated many different theories it the origin of the term Slavs, see also "Origin of the word Slavs":
late 13c., "person who is the chattel or property of another," from Old French esclave (13c.), from Medieval Latin Sclavus "slave" (source also of Italian schiavo, French esclave, Spanish esclavo), originally "Slav" (see Slav); so used in this secondary sense because of the many Slavs sold into slavery by conquering peoples.
This sense development arose in the consequence of the wars waged by Otto the Great and his successors against the Slavs, a great number of whom they took captive and sold into slavery. [Klein]
More common Old English words for slave were þeow (related to þeowian "to serve") and þræl (see thrall). The Slavic words for "slave" (Russian rab, Serbo-Croatian rob, Old Church Slavonic rabu) are from Old Slavic *orbu, from the PIE root *orbh- (also source of orphan), the ground sense of which seems to be "thing that changes allegiance" (in the case of the slave, from himself to his master). The Slavic word is also the source of robot.