Electric motors can be and are used in aviation - there's hundreds of them on any airliner you fly on.
As for using them for main propulsion, let's first look at the power/weight ratios. The highest number for Tesla's car motors is 8.5 kW/kg. The electric-aircraft-specific Emrax 268 delivers about 12 kW/kg.
In comparison, the Trent XWB delivers 430 kN of thrust at 300m/s flow rate, which equates to 64.5 MW of power, in a 7,550 kg package - a power/weight ratio of 8.5 kW/kg. However this isn't apples to oranges: this ratio is for the whole package, engine and fan, and measures useful output, like wheel power for a car.
In short, turbine engines are still lighter than electric motors, but the difference is not dramatic.
Where all-electric powerplants fail to stack up is range. I've elaborated on it in response to another question - Are there any hybrid electric planes?. The short version is that the maximum possible range of an electric aircraft is 10 nmi for every % of its weight dedicated to the battery. This limits the range of electric aircraft to 300-450 nmi, if sticking with the fuel fractions of known airliners.
But there are aviation applications where this is enough. The most important non-renewable resource consumed by modern aviation - the supply of 1960s Cessnas, without which no one could afford to become a pilot - will not last forever. Should authorities permit it, mass-produced Tesla powerplants could power trainers and GA planes at a fraction of the ownership cost of a certified avgas engine.