Would an analog mux be able to output various voltages?
Yes, you can do this, but there may (or may not) be a problem. You cannot use your selected voltage as a power supply. Look at figure 1 in the typical performance section. The effective resistance in the switch will be in the range of 60 to 90 ohms. And the maximum current you can pass is 20 mA. 

Even at 10 ma, a resistance of 60 ohms will cause a voltage drop of .6 volts, so in your example, if the load is drawing 10 mA, the output voltage will be 1.6 volts, rather than 2.2 volts.

This may not be a problem. For instance, if you are monitoring several power supplies with a high-impedance A/D converter, input impedance better than 10 k ohm, your measurement will be accurate to ~ 1%. But if you intend to actually power a circuit, you'll need to be very careful.