Can I install TheTruthSpy remotely?

No.

You need to have the device physically in your hand for about 5 minutes. Installation is simple. You simply open up a web page on the mobile and enter your code to download. The install begins automatically and setting the options is simple. You can also change the settings and control some aspects of the device itself, with secret remote SMS commands, or with user control panel website at http://my.thetruthspy.com

Note about iDevice (iPhone, iPad, iPod Touch): iDevice (iPhone, iPad, iPod Touch) installs are not downloaded this way, but installed through the Cydia utility on an already jailbroken iDevice (iPhone, iPad, iPod Touch).

Note about Android devices (Samsung,...): You can see the step-by-step installation guide here: http://thetruthspy.com/how-to-install-mobile-spy-on-android-with-thetruthspy/