No, the game is programmed to use the RTC inside the cartridge (see siirtc.c in pret's decompilation of Pokemon Emerald). The DS is designed on purpose to behave exactly like a GBA while it's playing GBA games, so Gen 3 games have no interface with the DS's separate RTC at all.
The symptoms you've described are consistent with the timestamp provided by Emerald's RTC getting stuck behind the timestamp written inside your save file. When this happens, most time-based events in the game will stay frozen even if the RTC is working, because the game only updates time-based events when the RTC counts ahead of the save file's timestamp (see clock.c in the decomp, more doco on the issue).
The most famous cause of this problem is replacing the internal battery, which interrupts the RTC's power supply (thereby resetting it) but does nothing to your save file. It's also caused if the RTC's power is interrupted for any other reason - it's entirely possible there was a spontaneous loss of power that reset the RTC and got you in this situation.
To fix it, you need to edit your save file or edit the RTC time to make sure the RTC counts ahead again. It can be done by dumping your Emerald save file (best way is using GBA Backup Tool on a DS flashcart) then using PKHeX to change the saved timestamp (SAV → Clock (RTC) → Time Elapsed).

^^ The culprit as seen in PKHeX for my game with a dead battery. If I were to replace the battery, I'd have to wait 953 days for the RTC to count ahead again. Or, I could modify this save to bring down the Time Elapsed - but even that won't last forever, as the battery will inevitably fail again.