The opamp's input common mode range doesn't reach close enough to the supply voltage to do this. It can reach to 0V but not to VCC.
You could use a rail to rail input opamp, but another problem remains if you want a GND-referenced output: the common mode rejection of the differential amp you built is dependent on resistor accuracy and matching, and thus not very good with 1% resistors.
If you can reference the output to VCC (for example, your ADC uses VCC as its reference) then you can use a rail to rail IO opamp, and you won't have common mode rejection problems.
simulate this circuit – Schematic created using CircuitLab
If you squint hard enough, this is a non-inverting opamp, with the usual gain of (1+R4/R3), except the input and output are referenced to VCC instead of GND. So,
Vout = Vcc + (1+R4/R3) * (V across R1 with the proper polarity ;)
If the range of your ADC is from 0V to VCC then it'll work just fine, but you do need a RRIO opamp.
If you want a ground-referenced output, then the best is to use a specialized current sense amplifier chip, which already contains everything you need. You can try MAX4376 for example. The reason I'm recommending this one is that it works with input common mode range down to 0V, which is not something that they all current sense amps do...