Galaxies consist of dark matter, stars, and gas. While gas is "collisional", i.e. it may interact hydrodynamically and cool, dark matter and, effectively, stars are collisionless. Hence, it is relative easy for an originally more or less spherical, gas-rich galaxy to collapse along the axis of rotation, while centrifugal forces prevents to collapse in the plane. In contrast, a spherical gas-poor galaxy will tend to keep its shape.
Spiral galaxies are blue
That is, a gas-rich galaxy will tend to settle down into a flat disk. Being gas-rich, such a galaxy will keep forming stars, and a spiral pattern will form.
Stars come in all sizes, but the most massive ones dominate the total luminosity (since $L\propto M^4$), and since massive stars shine with very energetic light, they are bluish/white.
Elliptical galaxies are yellow
Elliptical galaxies are the result of a merger between two or more galaxies. If the merging galaxies are very different in size, the smaller will just be "eaten" by the larger. But if the galaxies are similar-sized (and don't have too similar angular momenta), the event can be quite dramatic:
Tidal forces mess up the shape, gas is stripped from the galaxies, and collisions of gaseous clouds ignite massive starbursts. Such major mergers are quite efficient at suppressing angular momentum, allowing large amounts of gas to reach the galactic centers, feeding the central supermassive black hole leading to vigorous quasar activity (e.g. Hopkins et al. 2008).
Both the gas stripping, the radiation pressure from newborn, massive stars, the feedback from their fast and explosive deaths, and the active galactic nuclei, cause the merged galaxy to lose its gas. As the massive, blue stars die out, and with no gas to form new stars, only the smaller, yellowish/orange/reddish stars are left. The galaxy is said to be "quenched", ending up "red and dead".
Why not the opposite?
So the answer to your question is: The reason spiral galaxies are not red is that they keep forming new stars, the most luminous of which are blue.
And the reason elliptical galaxies are not blue is that they're unable to form new stars, so only the red ones are left.
Addendum thanks to Peter Erwin:
Spiral galaxies may eventually deplete most of their interstellar gas (partially through outflows) and stop forming stars. This cause them to gradually change color toward yellow. However, without the feedback from star formation, such a galaxy will also tend to lose their spiral structure, ending up instead as a lenticular (or "anaemic") galaxy (see e.g. Elmegreen et al. 2002).