Nothing would happen due to the ice caps melting. However, the increase in temperature that would be required to cause the ice caps to melt would result in an increase in pressure.
How much of an increase in pressure? Well, for a simplistic model, let's consider the ideal gas law: $PV = nRT$.
What this says is that, for a closed volume, the increase in gas pressure is proportional to the increase in temperature. This is temperature measured in Kelvin, so even a change of 10 C would lead to an increase in pressure of only around 100*(310/300 - 1) percent, or around 3%. (Based on an initial temperature estimate of around 30 C). Our atmosphere can also expand due to heat, so this would be an upper bound, not an absolute.
Melting both ice caps would lead to a sea level rise of around 70m, which would inundate many low lying areas and increase the surface area of the oceans, so we might have a bit more water in the atmosphere as well, leading to a slightly higher pressure, but the major change would be from the increased temperature.
Historically, large creatures are predicted to have flourished in an atmosphere which wasn't significantly higher pressure than ours, but rather had a higher proportion of oxygen in it. (Around 35% at max). This is hypothesized as the reason why arthropods may have grown so big, though the theory is somewhat contentions. The largest dinosaurs lived in a climate with an atmosphere similar to our modern one.