Allow access to RDS instance from EC2 instance on same VPC

To communicate outside of the VPC, each non-default subnet needs a routing table and an internet gateway associated to it (the default subnets get an external gateway and a routing table by default).

Depending on the way you have created public subnet in the VPC, you might need to explicitly add them additionally. Your VPC setup sounds like it matches Scenario 1 - a private cloud (VPC) with a single public subnet, and an Internet gateway to enable communication over the Internet from the AWS VPC documentation.

You will need to add an internet gateway to your VPC and inside the Public subnet's routing table assign 0.0.0.0/0 (default route) to go to the assigned internet gateway. There is a nice illustration of the exact network topology inside the documentation.

Also, for more information, you can check the VPC Internet Gateway AWS documentation. Unfortunately it's a little messy and a non-obvious gotcha.

For more details about connection issues, see also: Troubleshooting Connecting to Your Instance.

Since RDS requires you to have two availability zones when deploying in a VPC, you need to make sure that beanstalk is able to get to both of them via Network ACLs as well as the permissions for the instance based security groups.

Only your ELB and your NAT instance/NAT gateway need to be public subnets, everything else should be in private subnets.

Security groups are stateful and network groups are stateless so while you only need to allow inbound rules for the security groups, you need to allow BOTH inbound and outbound ports from your beanstalk subnet to both RDS subnets using Network ACLs. See Security in Your VPC.

Here is a sample eb create to create the beanstalk environment (replace square bracketed strings):

eb create [BEANSTALK_ENVIRONMENT] --instance_type m3.medium --branch_default --cname [BEANSTALK_CNAME] --database --database.engine postgres --database.version [x] --database.size 100 --database.instance db.m4.large --database.password xxxxxxxxx --database.username ebroot --instance_profile [BEANSTALK_EC2_IAM_PROFILE] --keyname [SSH_KEY_NAME] --platform "64bit Amazon Linux 2015.03 v1.3.0 running Ruby 2.2" --region us-east-1 --tags tag1=value1,tag2=value2 --tier webserver --verbose --sample --vpc.id [vpc-xxxxxx] --vpc.dbsubnets [subnet-db000001,subnet-db000002] --vpc.ec2subnets [subnet-ec200001] --vpc.elbsubnets [subnet-elb00001] --vpc.elbpublic --vpc.securitygroups [sg-00000001] --sample --timeout 3600

subnet-db000001 NETWORK ACL RULES:

Inbound: Port Range: 5432, Source [subnet-ec200001 (as ip range)], Allow
Outbound: Port Range: 5432, Source [subnet-ec200001 (as ip range)], Allow

subnet-db000002 NETWORK ACL RULES:

Inbound: Port Range: 5432, Source [subnet-ec200001 (as ip range)], Allow
Outbound: Port Range: 5432, Source [subnet-ec200001 (as ip range)], Allow

subnet-ec200001 NETWORK ACL RULES:

Inbound: Port Range: 5432, Source [subnet-db000001 (as ip range)], Allow
Inbound: Port Range: 5432, Source [subnet-db000002 (as ip range)], Allow
Outbound: Port Range: 5432, Source [subnet-db000001 (as ip range)], Allow
Outbound: Port Range: 5432, Source [subnet-db000002 (as ip range)], Allow

subnet-elb00001 NETWORK ACL RULES:

Inbound: Port Range: 80, Source 0.0.0.0/0, Allow
Inbound: Port Range: 443, Source 0.0.0.0/0, Allow
Outbound: Port Range: 80, Source 0.0.0.0/0, Allow
Outbound: Port Range: 443, Source 0.0.0.0/0, Allow

An additional note about Network ACLs -- many services don't respond on the original port but use an ephemeral port. So you may have to add the following to the inbound AND outbound network ACLs for subnets with EC2 instances:

Outbound: Port Range: 1024-65535, Source 0.0.0.0/0, Allow
Outbound: Port Range: 1024-65535, Source 0.0.0.0/0, Allow

There are also several useful scenarios in Recommended Network ACL Rules for Your VPC.

I hope this helps.