Q: Is this identity true?
A: Yes, Mathematica evaluates it as $$\sum_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}=\frac{\sqrt{\pi }\, \Gamma (x)}{\Gamma \left(x+\frac{1}{2}\right)}-\frac{1}{x},$$ which is another way to write the answer in the OP.
The identity holds for all real $x$ unequal to a negative integer. It holds in particular for $x=0$, when the sum equals $2\ln 2$.