Creating and Receiving Electromagnetic Field with RaspberryPi
Yes you can.  There's no reason you can't do what you are suggesting, but as Chuck said you will run into some issues.  

The biggest one I see is probably transmitting power.  You can only draw ~250 mA from the Arduino, so you'll have to overcome that limitation somehow (shouldn't be too bad).

The other issue is the stability of the transmitter waveform.  Most systems have a bucking coil (the GEM2 is an example; have a look for some papers, I think maybe by Fitterman, about the use of the Qcoil) about 1/3 of the way between the transmitter and receiver.  Without that, any system drift in both time and amplitude will be difficult to account for.  

I have no idea how accurate the timing or even what the digitization rate is on an Arduino.  Ideally you'd like to have several frequencies between say a few hundred Hz and maybe 50-100 kHz (if you were building a proper field instrument), but I think one frequency wherever you can get it as a proof of concept would still be impressive if it works.

There are a ton more issues I'm sure you'll run into, but these are the biggies I think.  I think Chuck has a great idea--start with a VLF receiver.  Get that working, and then you can use that as the receiver for your FEM system.

Your system will drift, and drift badly, I'm almost certain (even the commercial ones do).  Don't be discouraged.  If you can repeat measurements over the same spot, or even better construct yourself a calibration coil you can walk over, then you can characterize your drift and all will be right with the world.

I would LOVE to hear how this project goes.  Kudos to you for attempting this.  Please keep us updated and let us know if we can help.  

Andy (research geophysicist at the USGS and daily FEM instrument user)