Looking at http://www.serebii.net/xy/unobtainable.shtml, there are 27 Pokemon from Generation 3 that are listed as "Unobtainable" in X and Y. However, you can obtain a number of these through previous games.
If you own HG/SS, you can use the Pokewalker and the "Hoenn Starter" gift to get a majority of those 27 missing pokemon. You can catch Latias, Latios, Groudon, Kyogre, and Rayquaza in HG&SS as well, which takes care of a good chunk of the legendaries required to complete the National Dex.
The unobtainable list for HG/SS (http://www.serebii.net/heartgoldsoulsilver/unobtainable.shtml) lists only Regirock, Regice, Registeel, Jirachi and Deoxys as unobtainable, which means you can obtain 22 of the missing pokemon just from this game alone.
Using Black 2 and White 2, you could then catch Regirock, Registeel, and Regice, leaving Jirachi and Deoxys as the only two unobtainables.
This is where it gets tricky. If you participated in any of the events where Deoxys or Jirachi were handed out, then you would be able to bring the final two in from one of those games. However, if you missed out, you're going to have to wait until the next event where they hand out those pokemon. The only reason you would have to import them from Gen 3 would be if you had one of them on RSE (Like a Jirachi from the Pokemon Colosseum bonus disk). Otherwise, you could just wait for the next event, or get them from a previous event.
Hopefully, this will help you out in completing the National Pokedex. Good luck!
The OP's solution - originally posted in the comments:
The only way for the battery to resume time is to restart the game.
As much as I hated to do it, I restarted the game and now everything is working properly.
Best Answer
It's a change from SRAM (Static Random Access Memory) to FRAM (Ferroelectric RAM), EEPROM (Electrically Erasable Programmable Read-Only Memory), or other non-volatile memory (keeps its data even when it's not powered).
SRAM requires an active electric current to keep its memory active (called volatile memory). However, FRAM, NAND, and EEPROM memory are examples of non-volatile memory storage and do not require constant power to store their data. Improvements in non-volatile memory allowed the change to happen en-mass. The use of volatile memory for save states was greatest during the SNES/GB/GBC/Genesis/GameGear era, phased out during the N64/GBA/32X/Nomad era, and I don't think a single DS game used them. Gamecube obviously didn't.
The battery is still used for the RTC (Real Time Clock), which requires power to keep its clock source (an internal or external crystal oscillator) running, and updating of its time registers. Funny enough, modern computers still do this for their clock.
Maybe. Here's the thing. Pokemon games are constantly counterfeited. Fakes are common. Sometimes it's hard to tell. This site has very high res pictures of (supposedly) real cartridges.
For example, the LeafGreen has a Macronix MX29L010TC EEPROM. No Battery.
Yet Sapphire has the same EEPROM, but with battery.
Of course, Leaf Green and Fire Red do not use a RTC, hence the lack of a battery (while also causing certain things not to happen correctly, or taking longer, like berries growing).