How would membrane fluidity change if the phospholipid membrane had only one fatty acid?
Cell membrane doesn't have just phospholipid inside.
Components who influence fluidity in cell membrane are:
1) Fatty acids length 2) The grade of unsaturation of these fatty acids 3) Presence of cholesterol
So, if composed of just fatty acids, excluding every other component (carbohydrates, channels proteins and, mainly, cholesterol), the answer would be yes, its fluidity will change drastically.
This depends entirely on the fatty acid and the reference membrane.
Amphipathic membrane fluidity is based on the intermolecular forces between the hydrophobic parts of the membrane. Like olive oil vs. butter or margarine, the specific type of lipid determines the fluidity and the temperature at which a phase change occurs.
You can construct a synthetic lipid bilayer with several different kinds of saturated fatty acids (think butter) that would have a phase change at a higher temperature and be less fluid than one with only one unsaturated fatty acid (think an extract of a single type of fatty acid from olive oil) .
In fact, with regard to cholesterol (from the same reference):
cholesterol tends to make lipid bilayers less fluid, at the high concentrations found in most eucaryotic plasma membranes
The point here is that fluidity is not specifically about whether you have all fatty acids, one kind of fatty acid, some fatty acids and some cholesterol, or any proteins. You have to look at the type of each of these things, and, in particular, the intermolecular forces between the hydrophobic parts of your amphipathic molecules.