FMCW Radar: Need for Second stage FFT for velocity determination? I have understood that for the purpose of Range determination, the samples of the beat frequency are first assembled per chirp. The beat frequency is directly proportional to the range of the reflecting object.

So if there are N chirps, there will in a 2D matrix be N Rows of M samples each. When the FFT is performed along each row (Chirp), the spectrum analysis yields the value of the beat frequency of the corresponding chirp. Let us assume that the beat frequency is represented by Bin B.

If the object is stationary, then all of the rows will have a significant value and nearly the same value in Bin B [Magnitude and Phase].

If the object is slowly moving away and assuming that each bin represents a large range, then Bin B continues to remain the dominant bin with possibly reducing magnitude and increasing phase.

Question hence is: Why perform a second round of FFT along a column for determining velocity of a range bin? 

Why couldn't the individual phase of the FFT samples [Output of first round of FFT] along a column be taken and the velocity deduced from phase difference among bins of the column of a range bin?

After all, the phase of a received signal is proportional to the range of the object.
Actually the second round of FFT is exactly doing that, as by definition differentiation of phase is frequency. In realistic case if target is moving, the target position will not change but its phase will change along the column (also called slow-time, where the row in your case is called fast-time), the second FFT shows this changing phase as Doppler frequency. Furthermore, what second FFT also does it something similar to pulse integration. More the number of rows larger is your integration gain.