Is the seafloor of the Gulf of Mexico and Caribbean Sea mostly sedimentary rock?
Yes and no. Caribbean Sea and Gulf of Mexico have sedimentary materials and rocks at the top, but seabed sensu stricto is composed by sediments and not by rocks.

Oceanic Crust is composed by basalts and gabbros, both igneous rocks as you notice, and sedimentary materials above. We know that because of drillings and fossilized ophiolites:


  
  
  Source: earth.s.kanazawa


The Gulf of Mexico is not an exception and has sediments at the top, as said in this paper:


  "Most sediment originates from the adjacent land, primarily via fluvial transport. Direct precipitation of calcium carbonate and evaporate minerals takes place primarily on the Florida and Yucatán platforms and some coastal lagoons. The deep environments tend to be dominated by mud in a combination of terrigenous and biogenic sediments." (Davids Jr, R.A.,2017) 1


Caribbean Sea is neither an exception, as remarks Encyclopedia Britannica:


  "The ancient sediments overlying the seafloor of the Caribbean, as well as of the Gulf of Mexico, are about a half mile (about one kilometre) in thickness, with the upper strata representing sediments from the Mesozoic and Cenozoic eras (from about 252 million years ago to the present) and the lower strata presumably representing sediments of the Paleozoic and Mesozoic eras (from about 541 to 66 million years ago)" (Menzies,R.J. et al, 2019) 2


This article says there is one kilometre of sedimentary materials, enough to diagenesis to take place and so, there are sedimentary rocks under the unconsolided sediments.

At open oceans diagenesis may have still do not have place and there is only the ophiolitic sequence with thin sediments, but, as you can read in this related answer, Caribean Sea have old igneous rocks with a big stack of sediments above that have litified to sedimentary rocks (coming from continental rivers and carbonaceous platforms), becoming the biggest oil source zone on Earth after Arabian Gulf. 



1) Davis R.A. (2017) "Sediments of the Gulf of Mexico." In: Ward C. (eds) Habitats and Biota of the Gulf of Mexico: Before the Deepwater Horizon Oil Spill. Springer, New York, NY

2) Menzies, R.J., Ogden, J.C. (2019) "Caribbean Sea." In: Encyclopedia Britannica.