Minor nit: the OP asked how much current would flow into his antenna wire. His transmitter is rated at +0.6dbm, which is a POWER level, relative to 1 mW. The specification of Power Level doesn't , by itself specify the impedance.
Now somewhere else on the data sheet for the transmitter it will say something like RF Power was measured into an XXΩ load (usually 50 or 70Ω). The simplified calculation I did above was based on the bogus assumption that his antenna would be resonant, and that it presents something like a free-space resonant 1.2λ dipole feed-point impedance (72+j0)Ω to the transmitter, and that the transmitter will actually put +0.6dbm into that impedance.
If his antenna is actually a 1/4λ monopole operated against a non-existent or tiny ground plane (the PC board), then the feed-point impedance that the transmitter sees is 16 - j800, a very highZ, very capacitive.
Now the question becomes, how much power does the transmitter put into such a crappy load, (I'll be surprised if it is anywhere close to its rated power), and what is the actual current that flows into the antenna (it will be much, much less than the 4mA I calculated above).
The moral of this story is that a 1/4λ monopole without a proper 1/2λ by 1/2λ ground plane under it is a really crappy antenna!!!