$\Delta\Delta_v \Delta$ could be a counterexample. Here $v$ is a cut vertex and consider $<V(G_1) \cup \{v\}>$ be a graph of left two triangles which is not a block of $G$ because it is separable.

Consider delta symbol $\Delta$ as triangle. Now, suppose, there is a graph $G$ as $\Delta \Delta_v \Delta$ which means left $2$ triangles are connected with one triangle in the right side by a cut-vertex $v$ (removing $v$ disconnects the graph though one more cut vertex is there in graph $G$ as there is no restriction on the number of cut vertices in $G$ but $G$ must have a cut vertex labeled as $v$)

Now, suppose $G-v$ has one component, say, $G_1$ as $\Delta$/ which can be obtained by removing $v$, Now induced subgraph $<V(G_1) \cup \{v\}>$ will be $\Delta \Delta_v$ which is a subgraph of $G$ and it is separable by a cut vertex, say $u$ denoted as $\Delta_u \Delta_v$ and hence, not a block.