AWS Gateway API: Multi-Region deployment from the same domain

Unfortunately, this is not currently possible. The primarily blocker here is CloudFront. MikeD@AWS provides the info on their forums:

When you create a custom domain name it creates an associated CloudFront distribution for the domain name and CloudFront enforces global uniqueness on the domain name.

If a CloudFront distribution with the domain name already exists, then the CreateCloudFrontDistribution will fail and API Gateway will return an error without saving the domain name or allowing you to define it's associated API(s).

Thus, there is currently (Jun 29, 2016) no way to get API Gateway in multiple regions to handle the same domain name.

AWS has no update on providing the needful since confirming existence of an open feature request on July 4, 2016. AWS Form thread for updates