When designing a common emitter amplifier, one of the first thing you do is assume a value for the collector current (usually somewhere between 0.5mA and 5mA). Then use your equation, RC = (Vcc-Vc)/Ic, to calculate the value of RC which, at the chosen collector current, puts the collector at mid-supply voltage.
Next values for RE and the base bias resistors are chosen to give an emitter current which is equal to the required collector current. (we assume that Ie = Ic).
So it is effectively the base bias resistors and the emitter resistor that set the collector current, and if the collector resistor has been sized correctly then this collector current drops the correct voltage across the collector resistor to bias the collector voltage to mid-supply for equal possible output swing in either direction.
So collector current,Ic = (Vcc-Vc)/Rc which is approx. equal to the voltage at the emitter divided by the emitter resistance, Ic = Ve/RE