For the most part, yes, the ULN2003A will work in this circuit, but with some caveats.
The VCE(sat) is slightly higher for the ULN2003A vs the TIP120, which means the LEDs will have slightly less voltage across them (and so slightly dimmer). But that may not matter for your application.
My biggest concern is the total current through the ULN2003A chip when all three LEDs are on. Figure 14 of the datasheet shows the maximum current you can drive through each of the Darlington pairs. As you can see, if you have all three LEDs on at 100% duty cycle, each color can only draw a little more than 150mA. That's 450mA total. Any more than that and the chip will overheat. It's important to get a handle on how much current your LED strip pulls.
If, as you said in your comment, the strip is 1.5W with a 12V power supply, then presumably that translates to 125mA total. If that's true, you're probably ok then.
EDIT: You indicated you're using a 5V power supply. Since your LED strip is sized for 12V, the LEDs may not even light up. Especially the blue LED. If they do light up, they'll be significantly dimmer than they were designed to be.