That won't work because you're still trying to sink the 7 amps of current with the arduino and you're limited to 40mA. In electronics, there's very little difference between supplying "sourcing" or taking "sinking" current. It's so similar, the early electronics engineers guessed backwards, so physically, the electricity flows from ground to positive (they had a 50-50 chance and got it wrong when they set up the labelling conventions).
You still need the current to flow back to the battery, and it simply can't go through the chip. You need something else to handle the current, and what you'd use is a transistor. But you still need to be careful because 7 amps is such a large amount of power you'd need a big power transistor.
There's also a huge amount of heat involved. 7 amps at 5 volts is 35 watts. That's a lot of heat. Standard resistors are usually 1/4 watt or even 1/8 watt.
Hold everything, that schematic will make a mess of your Arduino. The full current will be sinked by the Digital IO pins. Those BLACK wires you show connecting the cathodes of the LED's will have the same current trhough them as the RED wires sourceing the LED's. You want a simple transistor switch to sink the LED cathodes to GND.