Perhaps I missed something here but, the Windows Update log shows you running a different computer than a Gateway desktop:
*********** Report: Initializing static reporting data ***********
2013-07-22 17:29:45:311 356 13f8 Report * OS Version = 6.1.7600.0.0.66304
2013-07-22 17:29:45:311 356 13f8 Report * OS Product Type = 0x00000003
2013-07-22 17:29:45:330 356 13f8 Report * Computer Brand = Acer
2013-07-22 17:29:45:330 356 13f8 Report * Computer Model = Aspire M5802/M3802
2013-07-22 17:29:45:334 356 13f8 Report * Bios Revision = P01-A0
2013-07-22 17:29:45:334 356 13f8 Report * Bios Name = Default System BIOS
2013-07-22 17:29:45:334 356 13f8 Report * Bios Release Date = 2009-08-18T00:00:00
******************************************************************************************************
Your report shows you ran the log on an Acer Aspire M5802/M3802 which is a desktop PC, not a Gateway desktop PC as you indicated in your OP. This shouldn't make a big difference, and maybe you have more than 1 PC at home, and you ran this report on the wrong PC??
I have never worked on an Acer desktop (in 30 yrs.), so perhaps they are not widely available here in the US, but I have worked on several Acer Aspire laptops, and they have plenty of issues. If I had direct experience with an Acer desktop, I could advise you to continue the repair. If your computer that actually has the problem really is a Gateway, I DO have plenty of experience with those, and I would advise you to continue trying to repair your Win7 windows installation.
However, based on what I've seen with the Aspire laptops, that does not inspire confidence in me that there is any chance at all of repairing your Windows installation. They have many problems with both their Motherboards and their hard drives. Your hard drive is in a machine that's probably around 4 yrs. old. Hard drives begin to fail as early as 2 yrs. in laptops and 3 yrs. or less in desktops. HAVE YOU RUN HARD DRIVE DIAGNOSTICS? Such as SeaTools for Seagate hard drives, or DLG for Western Digital drives? Some Aspire laptops I have worked on also come with built-in hardware diagnostics..have you run those on your hard drive? What was the result?
While you are taking time to run the hard drive test, you should also ensure that the RAM memory also passes your built-in manufacter-supplied hardware diagnostics.
If your hardware passes muster, per above testing I suggest you could certainly pursue a Windows Installation repair, if you have the time and expertise. For me, that can take 1-2 weeks, and I've been doing for 30 yrs. as I said. And they are not always successful, and I have no idea how accurate those will be on a desktop I've never seen before or worked on.
My suggestion, and it echoes Pauli's recommendations, is to fall back, backup all your data to an external hard drive, flash drive, cd/dvd disc(s), and reinstall your Windows7 from either a built-in Recovery Partition, or use the Recovery discs that came with your computer. If you really have a Gateway and not an Acer; there is a good chance you received those discs in the box your Gateway came in. If you have an Acer; not one of the Acer laptops I've worked on in that last year came with those discs, so you may have to look for a Windows Utility program, maybe until "ACER" or "MAINTENANCE" and create the disc set yourself. They will typically be from 2-4 discs for the Windows7 OS, and perhaps more discs for the "APPLICATIONS AND DRIVERS" disc as well.
Lastly, if you go the route that Pauli and I are suggesting to you, that should resolve your ability to upgrade your Windows7 to SP1.
I have done this on many Acer Aspire laptops and many, many Gateway desktops. Make sure that you select "ERASE ENTIRE HARD DRIVE PARTITION" when you do this, and not the "REPAIR WINDOWS PARTITION" option that Windows Recovery Console may present you. If you have file corruption or undetected viruses on your hard drive, that may not fix your problem. Reinstallation and Partition/Hard drive wiping fixes your SP1 update problem in better than 90% of the cases.
If it still does not work, you most likely have a faulty hardware component, hard drive, or Motherboard that needs to get replaced. At that point, you may want to take your Acer or Gateway PC to a local Certified Computer Professional to help get it repaired.
Looking forward to your responses back.
BIGBEARJEDI