I assume you mean to ask whether: $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$ For this question, the answer is no, set $\mathcal{E} = \mathcal{O}_X$. Choose $\mathcal{F}$ such that $H^i(X, \mathcal{F}) \neq 0$ for some $i > 0$. Then obviously $Ext^i_R(R, F) = 0$. However $Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(0)) = H^i(X, \mathcal{F}) \neq 0$.

I assume you mean to ask whether: $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$ For this question, the answer is no, set $\mathcal{E} = \mathcal{O}_X$. Choose $\mathcal{F}$ such that $H^i(X, \mathcal{F}) \neq 0$ for some $i > 0$. Then obviously $Ext^i_R(R, F) = 0$. However $Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(0)) = H^i(X, \mathcal{F}) \neq 0$.