The material would "probably" be shipped back to the USA, said California-based SpaceX.
The section, measuring about 32ft (10m) by 13ft (4m), was spotted between Bryher and Tresco on 26 November and brought ashore by boatmen.
Company representatives have travelled to Tresco to dispose of the wreckage.
"It's about being a responsible steward of our rockets after they are used," a spokesman said.
SpaceX previously confirmed the wreckage was from the first stage of a successful mission to the International Space Station from September last year.
"We don't need it for testing or analysis, it is a pretty straightforward piece of carbon fibre material," said the spokesman.
"There are no electronics or anything. We are just trying to clean up after ourselves."
The Receiver of Wreck Alison Kentuck said the debris officially belonged to SpaceX.
Many islanders wanted the debris added to the Valhalla collection of figureheads from Scilly shipwrecks in Tresco Abbey Garden
A spokesman for the UK's National Space Centre in Leicester said it had been "desperate" to keep the wreckage.
A section of a space rocket which was found off the Isles of Scilly, has been dismantled for disposal, its owners have confirmed.