I have a DC motor that has a stall speed current of 6 amps. Does that mean I should be supplying a constant current of 6 amps even when the motor is not experiencing a need for that much current?
Power supplies do not "push" current; components (like motors) "draw" current from the voltage source. For ohmic (resistive) loads, the current drawn equals voltage divided by resistance.
Voltage sources (power supplies, batteries, etc) do not have the ability to provide an infinite current. Internal resistance in the source will make the voltage "sag" as the current goes up, because voltage loss over any resistance (including internal resistance, wiring, etc) equals current times resistance.
For example, a small 9V battery has a very high internal resistance. (Another related term is voltage source impedance.) Draw 100 mA, and it may lose 2V. This means we can calculate the internal impedance to about 20 Ohms. A LiPo battery intended for high current discharge has a much lower internal resistance. Let's say you get a 1V drop when you draw 25 Amps. This means we can calculate the source impedance to 40 milli-Ohms. A bench power supply may be capable of increasing its internal source voltage to compensate for sag, until you reach the maximum current. Thus, a 12V supply may supply an even 12V all the way up to a rated max current -- let's call that 5A. After that point is reached, the supply will lower its voltage until the current draw is no more than 5A. (This is a "constant current" supply.) This typically works because the current draw of most loads is proportional to the voltage. However, for a "true" short, the supply needs to lower the voltage all the way to 0, because even at very low voltages (0.1V, say) more than 5A may be drawn by the "load" that's the short.
So, if you want the motor to be able to provide all the torque it is rated for, to the point where it stalls, at the voltage that the motor is rated for, then you have to use a power supply that is capable of supplying the rated voltage, at the rated stall current. However, you don't want to do that for any real time, as the motor will quickly overheat and damage itself.
No, you should be supplying the DC motor with a voltage that is sufficient for your requirements but the supply voltage should be capable of dispensing a current of 6A.
However, if you have a supply that is self-protecting by some kind of current limit circuit to (say) 4A, that is OK providing you realize that when starting a motor it is in a "stalled" condition and with current limited to 4A it may not turn the load you have attached due to stiction.