In a statement the Trust's chairman Sir Michael Lyons said he believed Thompson has "acted correctly throughout", claiming his decision was necessary to preserve the BBC's impartiality.
Thompson has also been asked to look at "wider lessons" to be drawn from the episode and to examine whether the BBC’s 37-year-old agreement with the DEC, is still appropriate.
Speaking in defence of Thompson, Lyons said it was impossible in the case of Gaza to separate the "political causes from their humanitarian consequences."
The BBC's refusal to run the appeal led to it receiving over 20,000 complaints from the public and more than 120 MPs who urged the BBC to show the appeal. There were also protests outside its headquarters in central London.
Sky News also refused to broadcast the appeal for the same reason as the BBC. However ITV, Channel 4 and Five all aired the two-minute piece.