Let $a,b,c$ be positive real numbers satisfying $c - a - b = 1 > 0$. After much manipulation, my problem reduces to evaluating the following limit: $$ \lim_{z\rightarrow 1^-} \left| \frac{1}{_2F_1(a,b,c;z) -\ _2F_1'(a,b,c;z)\sqrt{1-z}} \right|. $$ Here, a prime is a derivative with respect to $z$.
Now, we can use Euler's integral representation of Gauss' hypergeometric function to show $$ _2F'_1(a,b,c;z) = \frac{ab}{c}\ _2F_1(a+1,b+1,c+1;z). $$ Thus, while $_2F_1(a,b,c;1)$ converges by Gauss' hypergeometric theorem ($c - a - b > 0$), its derivative evaluated at $z=1$ diverges because $c + 1 - (a+1) - (b+1) = 0 \not> 0$. This is where I get stuck. I don't know how to deal with the denominator in this limit. I have a feeling this is quite obvious, but I can't see what to do. Any help would be appreciated.
The singularity of $\ _2F_1(a+1,b+1,c+1;z)$ at $z=1$ in your conditions is only logarithmical [see, for example, the Euler's integral formula] so $\sqrt{1-z}$ kills the contribution of the second term in the numerator to the limiting value.