Use 110 thermostat for 220?
You're looking for a relay which is designed for use in Electrical Code approved installations. 

First, do you want to upgrade to a safer and far more versatile 24V thermostat?  The sky is the limit with those, you can get programmable, touch screen, even Internet connected thermostats like the Nest.  They can also control heaters and even multi-stage.  They also let you place the thermostat where you please, and do the long wire runs with easy-to-install low voltage thermostat cable. 

If so, you'd want a 24V transformer to power the thermostat and remotely control the fan relay.  A 40VA transformer should be plenty, and these are readily available for about $12, as they are a common heating/cooling part. 

Then you can use an air conditioning relay (about $12 also).   This one works on 24V or 120V and mounts on a junction box knockout, saving you the trouble of mounting it in a junction box. 

Here's a combo transformer and relay which mounts on a junction box cover.  This keeps the 24V on the outside and the 120/240 on the inside, as required by Code.