Is it possible Private EC2 Instance send outbound traffic to Public ELB?

If you're accessing a Public load balancer that is in a different VPC (owned by you or by someone else) then the answer from Jhalak is correct - NAT Gateway is the way to go.

But I think your question is asking how you access the load balancer in the same VPC. You can still use NAT Gateway but there is a charge associated with that. Today, there's no simple way of getting a DNS record for the private IP of the load balancer. Instead, you might create a second private load balancer.

Also: Your question mentions ELB; please consider using ALB as it has many more features.

you can use NAT Gateway to send traffic out of your Private EC2/subnet.