The democratic process is an integral part of the republic. To be precise the US is a "democratically elected representative republic". Democracy has become a catch all term to refer to any form of government having a direct participation of the people in the election of their leaders.
To be "democratically elected" means that the people "en masse" directly decide what the rules of each election will be, as well as who is to be elected, and when. Once you do that you wind up with chaos - mob rule.
No one elected under these circumstances can truly be "representative" of the people since the people (the mob) can reelect another "representative" at their whim.
That's certainly what the Founders had in mind when they made this a Republic - not a Democracy.