Is this necessary for the current coming from the Arduino output pins?
Generally speaking: Yes. The ATMega chip data sheet says that you shouldn't connect anything that tries to draw more than 10 mA. It does not say that the ATmega safely limits the current to 10 mA. See Footnote.
recommended resistor rating
The current gain of the TIP122 is something over 1000 if you have a load that wants to draw 1A. Therefore something like 1 mA of base current will drive it into saturation (which is what you want).
My rule of thumb for this case would be to make the base current something on the order if 0.5 mA - 1 mA. So I might try a 4.7K Ohm resistor in series with the base. (Note that if your load draws less than 1A, the resistor can still bd 4.7K, but I might make it something like 10K if the load is 0.5A.) If the load is more than 1A, I would probably be a little more circumspect in my calculations.
For just about any garden-variety transistor such as the 2N2222, I have found that, with a collector current of a hundred mA or so, the gain will be something more than 100, so, it "just happens" that my target base current would, again, be something on the order of 1 mA. Then, again, the resistor that I might start with would be something like 4.7K Ohms.
Note that, in spite of the coincidence in these two examples, it is not
true that all transistor interfaces will turn out to require 4.7K Ohm base resistors. You should know the required load current and you should have at least an estimate of the current gain of the transistor at that value of collector current.
Even if it "seems to work" without a base resistor you should know that overstressing I/O voltages and currents can have cumulative bad effects that may shorten the lifetime of your CPU even if it doesn't fail immediately. Metal migration in the I/O circuitry of the IC will (probably) eventually cause a failure of, at least, that I/O cell, if not the whole chip.