Determine the approximation factor for greedy vertex cover algorithm

The approximation factor can be $Omega(log n)$.

Consider a bipartite graph $G$ with a set $S_L$ with $n$ nodes on the left side. Also consider a collection of sets $S_{R,1},S_{R,2},dots,S_{R,n}$ on the right side where each set $S_{R,i}$ has an edge to $i$ vertices in $S_{L}$ and no two vertices in $S_{R,i}$s have common edge . So $lfloor{frac{n}{i}}rfloor$ in it. So the sum of size of $S_{R,i}$s
$$sum_{i=0}^n |S_{R,i}|=Theta(nlog n).$$

After running GreedyVertexCover on this instance, it select all nodes on right side, but the optimal solution was to select all nodes in $S_L$.

Consequently, $$frac{ALG(G)}{OPT(G)}=frac{nlog n}{n}=log n.$$
So the approximation factor of your algorithm is $Omega(log n)$.