I HAVE FOUND THE ISSUE (I think).
The issue does not lie in the Liga III fixtures, but rather in the Liga II Relegation Playoffs. The game generated this fixture to be played on July 1st, 2030, which is AFTER the Season Update Day of June 30th. I think this fixture being unresolved caused all leagues not to update.
This fixture is usually played in late May, as can be seen in past seasons in the save; May 23rd/26th in 2029, May 24th/27th in 2028, May 26th/29th in 2027, etc.
In the initially uploaded file, the first leg of this tie was scheduled for May 19th between Dinamo Bucuresti and Unirea Slobozia, which has already passed without the game being played, but updates to June 5th and gets played if you deselect Liga III for removal and continue for a day.
HOWEVER, this fixture will not be played if you delay deselecting Liga III. It will try rescheduling the fixture, but it won't be played UNLESS Liga III is deselected for removal early enough.
I think the selecting for removal and deselecting for removal of Liga III causes some sort of refresh on the scheduling.
I have uploaded a save to the ownCloud titled "Romania Fixture issue save delayed enable to 26th" This file is on June 26th right after I have deselected Liga III for removal. Here the Liga II Relegation Playoff has updated to June 18th, but was not played.
Continuing forward reschedules the fixture to July 1st, but again does not get played and no fixtures are generated for any Romanian league as a result. This file was also uploaded as "Romania Fixture issue save delayed enable to 26th - July 1st"