Can an observer know what is the source of gravity? There's an observer in a closed room without windows under an influence of gravity force. Can he determine what is the source of gravity - whether it's a spinning motion, acceleration or huge mass object?
 A: Not 'locally', where 'locally' means doing experiments over a sufficiently small region.  Experiments done over larger regions can distinguish things.
What this means more formally is that spacetime is well-approximated by Minkowski spacetime in sufficiently small neighbourhoods, but can be distinguished from it on larger scales.
As an example, it would be very hard to distinguish sitting in a lift on Earth from sitting in an ordinary-sized lift being uniformly accelerated, and impossible to do so in the limit as the size of the lift goes to zero.  However if the lift was 100 miles on a side it would be much easier (measure the direction that freely-falling objects move in, relative to the lift, at one edge of the floor and compare it with that at the other, for instance).
A: LIGO is an example of an observer in a closed room.  LIGO has seen a gravitational signal deduced to be from two inspiralling black holes.  However, LIGO is non-local.  If LIGO were vanishingly small (ie: the arm lengths L were made infinitely small so as to make a truly local measurement) the mirror motion dx due to the gravitational strain would be too small to detect (dx=L* strain).  So, LIGO is not an example of determining the source of gravity from a "local" measurement, but does show it can be done by a measurement over an extended region.
A: I will touch upon a very practical (not theoretical) aspect of the question.
Let us take the case of gravity vs uniform acceleration of a man sized lift, say between 4 and 10 feet tall, 4 feet wide and 4 feet deep. Let us ignore the spinning motion for this example
The assumption is uniform acceleration. However, uniform acceleration means a constant force at all the times. Think of any practical way of providing a constant force. Practicality is the key here.
Now, put an object on a sensitive weighing scale, and jump on the floor of the lift. In case of a constant force, there will be a momentary change in the weight of the object. In case of the gravity, either that change will not be there, or it will be much less than that of the constant force.
So, by bouncing inside the thing, it would be possible to figure out what it is.
In order for the lift to give same effect in case of a jump inside, it has to be as heavy as the planet. But in that case, It will cause enough gravity anyway. 
