How to use elastic IPs without exposing traffic to the internet

Elastic IPs are public by nature.
Is there any possible way of leveraging Elastic IPs for internal load balancing on services deployed on private subnets, without routing the traffic to the public internet?
For instance, if I deploy may own MariaDB cluster with multiple nodes, as a backend DB for HDP services (Hive, Oozie, Ranger, etc), I don't want the queries issues by those HDP services to be routed through the internet. I want all the communication between HDP services and MariaDB to be private to my own VPC.
Is there any way to accomplish this?
Thanks,
Julius