The previous answers correspond to the recursive model of the form $$[x_n,y_n]=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1})]$$. However, the state-space model of the question is of the form 
\begin{align}
[x_n,y_n]&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},x_n)]\\
&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},f(x_{n-1},y_{n-1}))]
\end{align}
The latter equation is the correct form that should be applied with the methods proposed by the previous answers. Another method could be to use `NestList`

 h[{x_, y_}] := {x + 1/2 (Sqrt[1 - y^2] - x), 
 y + 1/2 (Sqrt[1 - (x + 1/2 (Sqrt[1 - y^2] - x))^2] - y)};
 NestList[h, {N[0 + 1/2 (Sqrt[1 - 0^2] - 0)], 
 N[0 + 1/2 (Sqrt[1 - (0 + 1/2 (Sqrt[1 - 0^2] - 0))^2] - 0)]}, 10]

> {{0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 
> 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}, {0.780556, 
> 0.624467}, {0.780804, 0.624622}, {0.780865, 0.624661}, {0.780881, 
> 0.62467}, {0.780885, 0.624673}, {0.780886, 0.624673}}