Yes, and no.
Theodore usually represents an Anglicized version of Greek
Theodoros, but you can't assume that every
Theodore does so, some may indeed be meant as anglicized versions of
Theodoric.
Theodoric however is derived from the Helleno-Latin version of a Gothic name probably more like
Thiudreiks (the -s is the Gothic form of the earlier nominative case indicator -az, dropped entirely in most other recorded Germanic languages except for a few very early inscriptions). The Helleno-Latin form
Theodoric (the Empire being dominated by the eastern Byzantine court at the time) is undoubtedly influenced by Greek
Theodoros, in the same way that Gothic Athnareiks is recorded as
Athanaric under the influence of the more familiar Greek
Athanasios. It's a process called "folk-etymology". English Theodric, and
German Dietrich or Tietrich are cognate with
Theodoric and
Thiudreiks (i.e. they share a common nativity or origin), one is not derived from the other (Tedric and Didric are probably anglicized versions of early continental forms).