No, it is not possible. Elastic IPs are tied to your account, not to individual instances, so the instance's interface IP will always be private.
EC2 Private IP Address: The internal RFC 1918 address of an instance that is only routable within the EC2 Cloud. Network traffic originating outside the EC2 network cannot route to this IP, and must use the Public IP or Elastic IP Address mapped to the instance.
EC2 Public IP Address: Internet routable IP address assigned by the system for all instances. Traffic routed to the Public IP is translated via 1:1 Network Address Translation (NAT) and forwarded to the Private IP address of an instance. The mapping of a Public IP to Private IP of an instance is the default launch configuration for all instance types. Public IP Addresses are no longer usable upon instance termination.
EC2 Elastic IP Address: Internet routable IP address allocated to an AWS EC2 account. Similar to EC2 Public Address, 1:1 NAT is used to map Elastic IP Addresses with their associated Private IP addresses. Unlike a standard EC2 Public IP Address, Elastic IP Addresses are allocated to accounts and can be remapped to other instances when desired.
Source: this article on elastic IPs.
If this really is a deal-breaking feature, Rackspace next gen cloud servers gives you a public IP on eth0 by default.