If the universe is expanding at an accelerating rate, such that the galaxies' moving away from each other is accelerated, then time should also slow down. And when universe will accelerate to the speed of light, then time should stop.
Does time slow down because universe is expanding at an accelerating rate? Can any physicist explains if this is true or not?
Yes, time does run slower for far-away objects, as observed from our point of view; this is a prediction of general relativity. And yes, because expansion accelerates, this time dilation slowly, very slowly, becomes more pronounced (this would happen even if the expansion didn't accelerate, but just continued at the same rate).
This time dilation is a well-known effect, and is always taken into account when doing observations. For instance, when observing distant supernovae, one is often interested in how their luminosities decrease as a function of time. This is called their lightcurve. In order to compare lightcurves at different redshifts, they are usually converted to their restframe, i.e. how they would look if you were "standing next to the supernova" (e.g. Goldhaber et al 2001).
However, time dilation does not work exactly as you seem to think. At galaxy at a redshift of $z$ has its time dilated by a factor of $1+z$, so time runs twice as slow for a galaxy at, say, $z=3$ than for a galaxy at $z=1$. Galaxies with redshifts larger than $z\sim1.5$ recede faster than the speed of light, and time does not at all stop here. Only for $z\rightarrow\infty$, i.e. at the beginning of time at Big Bang, does the time dilation approach infinity.
In general, better refrain from using 'inflation' unless you're talking about the early Universe-inflation. Rather, the Universe is expanding at an accelerated rate.
The crux of this question seems to be in what you call 'the' time. There is no such thing as 'the' time, and I'm not sure what definition of time you're expecting to stop. Time is perfectly well defined in any restframe and will not be affected by the overall expansion of the Universe, at whatever rate.
Our time does not slow down due to the expansion of the universe. And time will not stop.
However time dilation does occur, and this is something different.
When something is moving at a very high velocity, relative to us, we see time slowing down for it. As more distant galaxies are moving away from us at great speeds, events in those galaxies would (from our point of view) appear to run slower. This is observed in the at which supernovae brighten and fade described in this paper.
The inhabitants of those galaxies would not perceive any change in the rate of time. (rather they would see our clocks slow down, as we are the ones moving fast, from their perspective)
The rate of expansion is increasing (though why this should be is mysterious) but no part of the universe will accelerate to the speed of light, as nothing may travel at the speed of light. So time dilation will remain finite.
Thank you for your interest in this question.
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