I would like to put a 2-speed Shimano road shifter on my bike.
The bike is 3-speed front crankset. Could I put these two together?
If your shifter is a 2 position brifter (ie an STI lever) then it will only be in the right place on one chainring. The second position may be acceptable, or it may be off enough to create clatter. The third chainring will be unavailable to you completely.
If your shifter is a friction shifter, then the entire set of chainrings will be reachable and tuneable assuming you have sufficient range in the front derailleur.
I successfully shifted on a quad chainring using a thumb friction shifter on a custom bike. You could also look at using a bar-end shifter in friction mode.
Since you mention road bike, its going on drop bars, and STI / brifters is where its at for roadies. You'd be best off getting a triple-position left brifter (used shouldn't be too expensive)
The other option is to look into a compact double crankset, which will let you use a smaller inner chainring, paired with a wider range cassette at the back could approach or even equal a 1:1 ratio with an 11-32 cassette and a 32+48 chainring setup.
Depending on the age of your shifter, it may have a pseudo third position which is the trim setting. This lets you do a sub-shift on the front mech so to avoid the chain clattering away on the cage as you ride in a sub-optimal gearing like big-big, or less often in small-small. This Trim position should not be confused with a triple shifter.... when in doubt check the web for the part number.
Generally speaking a triple anything ends in a 3 in the Shimano world.