No. There is not an operation like vector / vector.
By the way, pressure in general is NOT a scalar, but a Tensor:
press = {
{pxx, pxy, pxz},
{pyx, pyy, pyz},
{pzx, pzy, pzz}
};
and the force experienced by a surface ( given by a vector surf = { sx,sy, sz } is given by
force = press.surf
Often the off-diagonal elements of press are 0 (this is most likely not the case in vsicous fluids) , and if in addition the diagonal elements are equal = p ( an isotropic medium) the force points in the direction of surf. Then you can write
p = Norm[ force ] / Norm[ surf ]
press = {
{p, 0, 0},
{0, p, 0},
{0, 0, p}
};
f1 = press.{1, 1, 1}
f2 = press.{2, 3, 4}
Norm[f1]/Norm[{1, 1, 1}]
Norm[f2]/Norm[{2, 3, 4}]