Do "Devi" and "Devil" have related roots?

To answer this question, I start with the related Oxford English Dictionary entries. Devil has a long etymology, of which these two excerpts cut close to the source:

ultimately < Hellenistic Greek διάβολος (in Jewish and Christian use) the Devil, Satan (Septuagint, New Testament), specific use of ancient Greek διάβολος accuser, slanderer, use as noun of διάβολος backbiting ( < δια- dia- prefix1 + -βολος (see peribolos n.), after διαβάλλειν to slander, lit. ‘to throw across’),

...

With reference to the adversary of the Judaeo-Christian God (see sense 1), Hellenistic Greek ὁ διάβολος (with definite article) is used in the Septuagint to translate biblical Hebrew ha-śaṭan , lit. ‘the adversary’ ( < ha- the + śaṭan Satan n.); early Latin versions of these texts (Vetus Latina) usually translate this as diabolus , whereas the Old Testament of the Vulgate uses Satan in most passages (except originally in Psalm 108:6, as it incorporates the Gallican Psalter based on the Septuagint; compare quot. a1382 at sense 1).

The Greek New Testament uses both ὁ διάβολος and ὁ Σατανᾶς ( < Hebrew: see Satanas n.) to denote the adversary of God.

(Hellenistic Greek διάβολος also occasionally appears in biblical contexts in its more literal sense ‘accuser, slanderer’ (Septuagint: Esther; New Testament: Timothy, Titus).)

Deva is relatively simple:

Sanskrit dēva a god, originally ‘a bright or shining one’ < *div- to shine.

So deva is rooted in Sanskrit, and devil in Greek. The Sanskrit root goes back to "to shine"; the Greek root goes back to "to slander." At least so far, there is no connection between the two.

If you look up the Indo-European roots in the American Heritage Dictionary, you find Deva instead grouped with another kind of divinity. Under "dyeu-" we have everything from deus (God) and diva to Deva and words derived from the Latin dies (day): diet, diary, and so on. Devil has another root entirely, "gwelə-." So I'd look there for a starting point on what words are related to Deva and devil.