No, a writer cannot approve a post. By the set of permissions defined, a writer can only perform tasks of creating a post - add text, images and a Schedule to it.
The post once saved by the writer moves to the Approval List under Planner. Here, a writer can make changes to the post such as edit or delete it. However, the posts can only be approved by the team member who has the access to Approve Posts.
In case a post is not approved, it will stay in Approval List.