Why won't my A/C turn on after I replaced the thermostat batteries?
Sounds like you may not have power from the transformer. You can check this using a multimeter, by testing the voltage between the red R wire and the blue C wire.  You should read somewhere around 24VAC, though depending on the system it could be anywhere from about 6VAC to 30VAC.

Most thermostats only use the batteries to power the thermostat itself, LED display, programming, date/time, etc.  Thermostats with a C wire (like yours seems to have), sometimes only use the batteries to remember your settings during a power outage.  The actual power for signalling heat/cool, is supplied by a transformer in the furnace itself (the red R wire). If there is no power on the R wire, the thermostat has no way to call for heat/cool.  In this case you'll have to have an HVAC tech troubleshoot, and possibly replace the transformer.

You could use your multimeter to verify the furnace itself is getting power, though this is slightly more dangerous since you'll have to open the service panel on the furnace. This could put you in danger of electrocution, and should be avoided if you are not familiar with HVAC systems.

The only other thing I can think for you to check, is to look for a serviceman switch and make sure it's in the ON position.  A serviceman switch is a switch on the feed line to the furnace, that allows servicemen to turn the power off to the furnace to service it.  It should be located very close to the furnace, and within line of site of the furnaces service panel. If there is line of sight from the furnace service panel to the breaker panel that feed the furnace; or this is an old installation, you may not have a serviceman's switch.

Your best bet, is to simply call in an HVAC technician to service the furnace.