You said the 24 volt wire connects to one side of the contactor. Where does the other 24 volt wire connect to? Is there a pressure switch or low ambient control in your a/c, wired in series? You may have two seperate problems. If you measure 24 volts across the 2 wires entering the outdoor unit, then the contactor should pull in. If it doesn't, then there is another problem in that a/c. (pressure switch open, low ambient control stuck open ect..)
Is this replacement transformer rated for 208 volts and 230 volts? Did you match up the VA rating? Is there 4 or 6 wires on this transformer? You may be mis-wiring the primary side if you have 6 wires. Most universal transformers have the 2 wires for 24 volts, and 4 wires for the primary, but you only use 2 of them. The common is black for 120 volts, 208 volts and 240 volts. Use the white and black for 120 volts, the black and red for 208 volts and the black and orange for 240 volts.
It's unlikely a 24 volt short can cause the primary side to fail, but I would still wire a 3 amp fuse on the 24 volt load side of the transformer, and see if it blows. If it does, then test the 24 volt 2 wire for continuity, and make sure there is no continuity when the wires are seperated.
If you disconnect the 24 volt wiring from the transformer, will it still burn out? Bring a new contactor down to the transformer, wire the 24 volt wires directly to the contactor coil and see if it pulls in. That way you eliminate the low voltage wiring feeding the outdoor unit, and the controls inside.
Also if you manually push the contactor coil in, at the outdoor unit, does the compressor and fan start?
Let me know what you find.