Let
$$M = \begin{pmatrix} A & B \\C & D\end{pmatrix}$$
be a block matrix. Define $a_n:=n^t A n$ for some real vector $n$ and analogously $b_n, c_n, d_n$. Does $\Re(x^*Mx) > 0$ for all $x \neq0$ imply the invertibility of $$\begin{pmatrix} a_n & b_n\\ c_n & d_n\end{pmatrix}?$$
I'm assuming that your $*$ means Hermitian conjugate (= conjugate transpose), and $n \ne 0$.
Note that for scalars $r$ and $s$, $(\overline{r} n^t\ \overline{s} n^t) M \pmatrix{rn \cr sn} = r \overline{r} a_n + \overline{r} s b_n + r \overline{s} c_n + s \overline{s} d_n = (\overline{r} \ \overline{s}) \pmatrix{a_n & b_n\cr c_n & d_n\cr} \pmatrix{r\cr s\cr}$. If $\pmatrix{a_n & b_n\cr c_n & d_n\cr}$ was not invertible, you could take $\pmatrix{r\cr s\cr}$ to be a vector in its null space and get 0.