Looking at this nice computer, wondering if it can handle minecraft and other games on high settings?

no. weak cpu, barely even a gpu in that thing. i strongly recommend not buying this.

the computer that can run gta4 on highest hasnt been invented yet.

yes, it has. Show me where you got that info from.. And also OP I'd recomend building you own PC, much cheaper than buying one from OEM and you can overclock.

This was my first DIY computer, but mine came with a GTS 250 and an e5400. I believe its around $500, but its a little dated. I could play GTA 4 on medium settings with a GTS 250 256 mb (or 1 GB, dont remember) so you should be able to live with that. But honestly, its not that hard to build a computer. The hardest part for me was trying to find out where to plug in the damned case power button xD. Theres a first time for everything

If you have even modest manual dexterity and eyesight (like me), you'll have no trouble building a PC. Most parts fit only one way, and there are lots of YouTube videos and other [illustrated] online guides on how to do it.
Take a look at any of the recent $500-$650 SBM builds for an idea of what can be done.

http://secure.newegg.com/Shopping/ShoppingCart.aspx?Submit=view

Here is a decent build

Other people can't see that besides you. use pcpartpicker.

Here's an Intel-based build that is within your budget and will meet your goals.

EVGA 01G-P4-3655-KR GeForce GTX 650 Ti BOOST 1GB 192-bit GDDR5 PCI Express 3.0 x16 SLI Support Video Card
Model #:01G-P4-3655-KR
Item #:N82E16814130914
$149.99 $149.99
Can play any game, typically on "High" settings.

Western Digital WD Blue WD10EZEX 1TB 7200 RPM 64MB Cache SATA 6.0Gb/s 3.5" Internal Hard Drive - OEM
Model #:WD10EZEX
Item #:N82E16822236339
$74.99 -$5.00 Instant $69.99
Nothing special; typical consumer drive. I've had good results with WD drives.

ASUS P8H77-M LE LGA 1155 Intel H77 HDMI SATA 6Gb/s USB 3.0 uATX Intel Motherboard with UEFI BIOS
Model #: P8H77-M LE
Item #:N82E16813131964
$79.99 -$8.00 Instant $71.99
Going for a budget board here, without giving up quality. There are a few other choices around this price, but not many.

LITE-ON DVD Burner - Bulk Black SATA Model iHAS124-04 - OEM
Model #:iHAS124-04
Item #:N82E16827106289
$17.99 $17.99
One optical drive is much like another.

Intel Core i3-3240 Ivy Bridge 3.4GHz LGA 1155 55W Dual-Core Desktop Processor ...
Model #:BX80637i33240
Item #:N82E16819116773
$124.99 $124.99
This hyperthreaded i3 should do well on even the latest games, with the possible exception of those few that are very heavily threaded; even then it shouldn't be "bad."

Rosewill CAPSTONE-450-M 450W Continuous @ 50°C, Intel Haswell Ready, 80 PLUS GOLD, ATX12V v2.31 & EPS12V v2.92, SLI/CrossFire ...
Model #:CAPSTONE-450-M
Item #:N82E16817182261
$99.99 -$30.00 Instant $69.99
For $10 more, you can get a Seasonic 550W unit, but on a budget this PSU (and the platform on which it is based) has reviewed very well. I'd buy one.

Rosewill CHALLENGER-U3 Black Gaming ATX Mid Tower Computer Case ,comes with Three Fans-1x Front Blue LED 120mm Fan, 1x Top ...
Model #:CHALLENGER-U3
Item #:N82E16811147060
$69.99 -$10.00 Instant $59.99
I like Rosewill cases. Among their advantages, they tend to come with more fans than nearby competitors like CM.

G.SKILL NS Series 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10600) Desktop Memory Model F3-1333C9D-8GNS
Model #:F3-1333C9D-8GNS
Item #:N82E16820231643
$61.99 $61.99
RAM is getting pricey! I've always had good results from G.Skill.

Subtotal: $626.92
Add Windows 7 for $80 (with a promo code) and you'll have a decent, upgradable system able to do well in modern games. At the high end of your budget, but does not go over (assuming you have a prior keyboard and mouse).

You could put an i5 in the build I posted.

Those are all Newegg.com prices. If you are not in the USA, please post where you are and where you can shop.

I believe it will handle most games on high settings; older and undemanding games (e.g. Minecraft) on Ultra/Max settings.
It will be a fast PC, but if you want "lightning" fast, you'll need to add a SSD (that won't help FPS, but load times of even hogs like MS Word will be rapid). You can always do that later though.

oops xD