The system requirements for 10.5.x don't mention Oracle 12.2 or 13.x. It's unlikely that Esri will announce a time when they will be supported. Contacting Tech Support to get on a list is your best recourse.
After aplying the patch, I tried to connect with ArcGIS 10.5.1 Desktop to Oracle 12.2 64bits (Enterprise Linux - Oracle Developer days virtual machine).
The ArcGIS 10.5.1 Desktop create the database connection but when I select it to connect, the ArcGIS Desktop blocks and stays Not Responding.
I used several oracle client (and instant) both x64 and x32 (12.2 and 220.127.116.11 as it says on patch)
I can connect through sqldeveloper and sql plus easy connect.
This is probably something to discuss directly with Tech Support. The alternative is to post a new question, giving lots of details on the exact OS and software, the exact PATH contents, and the exact procedure to produce the error.
What is the exact version of the Oracle database that you are trying to connect to. You mention 12.2, but the ESRI patch documentation explicitly references 18.104.22.168. Could it be that you are at a higher version of Oracle than 22.214.171.124?
The release is 126.96.36.199, the same as the Patch.
I was trying connecting with system user.
I created a new user, and now I can connect to the Oracle, but the connection took 5 to 10 min to connect!
I just created one table to see it on ArcGIS 10.5.1 Desktop.
Is the system user a user with elevated privileges to the database whereas a new user just has viewer privileges? I'm asking because it seems odd that you could not make a connection with the system user.
it seems some kind of priviliges that ArcGIS is trying to access.
With a simple user with grant connect session took 5 min.
With system user took more than 1 hour.
With sys user is immediatly.
There are very few reasons to ever connect using ArcGIS as SYSTEM (it probably qualifies as a Worst Practice). This is especially true if the number of objects in the database is large, since connection time closely correlates to the number of objects visible by the user (and SYSTEM can see everything, including the thousands of demo/sample tables that should not reside in any production server).