If you kept the working distance the same and only change the power of the laser, does the spot size change? So is the spot size independant of power?
The point size is independent of the power, but since the impact point of every photon is subject to the uncertainty principle your point which is the sum of many photons does not have completely sharp edges but rather a gaussian distribution over an area.
Since the brightness decreases from the center because less photons hit there the formerly dimmer parts on the edges of the point become brighter if you increase the power, so with your eyes or sensors you will experience the more powerfull spot not only brighter but also larger since now you also see the parts of the area illuminated which where also illuminated, but to dim to see, when you set the laser with lower power.
If you define the point in a way that in order to be a point a certain number of photons must land inside its specified area this area will indeed increase if you increase the power, but the edges will always stay unsharp.
Short answer is in principle yes, but...
I assume that you are pointing the laser on a pixelized photo detector and you are not focusing on the detector. Also I assume that you are using a normal TEM00 laser mode and nothing freaky/ugly.
The typical cross section shape of a laser profile is a Gaussian. So the answer depends on:
The measurement Depending on how you define the spot size you might get different results. If you are fitting a Gaussian to your profile and extract $\sigma$ or if you calculate the FWHM, then changing the power will not change the measured spot size. But if you for example just threshold the 2D Profile image and then calculate the distance from that, you might get different results. This points are only valid if you are in the linear range of your sensor. If you are exposing the non-linearities of the sensor things will start to go wrong.
Laser source Depending on your laser source and what you so with it afterwards, the Gaussian approximation might not be a good approximation. There are processes which might change your profile slope with increasing power. For example your profile can start to look like a flat top. This can interfere with your chosen profile evaluation.
There are some nonlinear effects that can change the spot size when power increases, such as filament propagation of laser radiation, say, in air (https://en.wikipedia.org/wiki/Filament_propagation)