Tsunami speed variations in the deep ocean caused by seawater density stratification is investigated using a newly developed propagator matrix method that is applicable to seawater with depth-variable sound speeds and density gradients. For a 4 km deep ocean, the total tsunami speed reduction is 0.44% compared with incompressible homogeneous seawater; two thirds of the reduction is due to elastic energy stored in the water and one third is due to water density stratification mainly by hydrostatic compression. Tsunami speeds are computed for global ocean density and sound speed profiles, and characteristic structures are discussed. Tsunami speed reductions are proportional to ocean depth with small variations, except in warm Mediterranean seas. The impacts of seawater compressibility and the elasticity effect of the solid earth on tsunami traveltime should be included for precise modeling of transoceanic tsunamis.