Can I use my wireless keyboard receiver to connect to Internet via my Wi-Fi router?

I have a desktop computer with Windows 7 and I recently purchased a Logitech wireless keyboard that has an "Unifying" transceiver. Can I use it to get Internet on my desktop by connecting to my home router?

No you cannot - short direct answer.

More of the technicality of why it cannot - Most wireless keyboards / mice use 2.4 GHz radio frequency for all their wireless communication. Though Wifi technically uses the same 2.4 GHz frequency but the signals confirm to one of the set IEEE standards such as 802.11b/g/n.

So, while the wireless trans-receiver can technically receive the signal but due to the way it's hardware will be built, it will ignore any of the Wifi signals.

The answer for your question is no. Logitech's unifying transceiver is used to connect the logitech's unifying devices like mice and keyboards and you can't use the same for your desktops internet connection.