Once we have installed and configured CA Access Gateway R12.8, we cannot start it when running sps-ctl script. We get the following errors:
(13)Permission denied: AH00072: make_sock: could not bind to address [::]:80 
(13)Permission denied: AH00072: make_sock: could not bind to address 0.0.0.0:80 
no listening sockets available, shutting down 
AH00015 --> unable to open logs.

We do not have this port in use. Why is this error appearing? How can we solve it?

Access Gateway R12.8 on RHEL7.4 x64

In RHEL, ports from 0 to 1024 are reserved for system use only. This means that in order to use one of them, you must have the authority to change/access basic system settings. The root user has such privileges and can actually use a port from the range 0 - 1024. 

In your problem, the system response indicates the root of the problem : 
(13)Permission denied: AH00072: make_sock: could not bind to address [::]:80 
(13)Permission denied: AH00072: make_sock: could not bind to address 0.0.0.0:80 
no listening sockets available, shutting down 

When the http daemon starts, it tries to bind the 80 port, which is a port within the system assigned ports and as such it can only be accessed by root. 

In order to be able to run it as a non-root user you could change the port to a non-reserved one (higher than 1024), or use an account with enough privileges.