Can Aurora PostgreSQL be used with AWS AppSync?

You can use the AWS Lambda resolver available in AWS AppSync to access Aurora Postgres. The code is similar to how you would access a relational database using any language. For example, you could use node-postgres with NodeJS to implement the Lambda function.

It can not. You can use Aurora Serverless as the data source which is driven by Data API (still in beta), this allows you to configure resolvers as database queries. That being said Data API is still very slow and Aurora Serverless has a cold-start of 30sec or so as it needs to run from VPC. I would recommend avoiding production but worth playing around.

You are much better of using Lambdas as resolvers or running HTTP RestFul calls from within the resolvers.

Ignore the comments provided in the answer, no disrespect, but the comments are coming from people who never managed production at scale. The fact you have a fully manage GraphQL service at scale as well as with high-security posture will save you months of maintenance nightmares when your product(s) will reach anything close to 1MIL revenue.