How to access private S3 bucket using VPC gateway endpoint?

Block public access must be disabled to access by object URL.
Only via VPC endpoints if the following bucket policy is set.
However, it has not been verified whether access by object URL is possible.

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Sid": "AllowVPCe",
            "Effect": "Allow",
            "Principal": "*",
            "Action": [
                 "s3:Get*",
                 "s3:List*"
            ],
            "Resource": [
                "arn:aws:s3:::BUCKET-NAME",
                "arn:aws:s3::: BUCKET-NAME /*"
            ],
            "Condition": {
                "StringEquals": {
                    "aws:sourceVpce": "VPC Endpoint ID"
                }
            }
        }
    ]
}

You only have to disable Block Public Access if you want to access your bucket as a static website (not recommended). Otherwise if you're accessing it via its REST interface (e.g. via the AWS CLI or API) then you just need the right credentials. As you said, an IAM Role allowing your EC2 instance to access S3 should be fine.

To use a VPC Gateway endpoint, you need an entry in your subnet route table directing an AWS-managed Prefix List to that gateway.