Q:

Can it be said that the tenets of democracy are philosophical in nature? Why/ why not?

Philosophy is defined as the study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline. Different philosophies examine how we should live our lives, including how we should structure our laws.
Democracy, on the other hand is defined as a system of government by the whole population or all the eligible members of a state, typically through elected representatives.
Do the actions of the populace in agreeing to live (usually) peaceably in a democratic society and the problems associated with the system count as a type of philosophical statement (even if it's not a conscious choice of the people so involved)  

A:

Your intuition that the core of democracy is an unspoken, unwritten, un-examined social contract is mostly right. While there are non social contract theories, e.g., instrumental (or utilitarian) arguments for democracy, many scholarly arguments for democracy are guided by the social contract intuition. But there are many written, examined works.
Rousseau, for one, argued for democracy from the social contract perspective. To Rousseau, a legitimate sovereignty must represent the general will, and people would sign on only democracy since democracy is the most likely to be able to do so. 
The general will theory of democracy became out of favor among scholars due to the findings from the social choice theories (specifically, Arrow's Impossibility Theorem). Political scientists and political philosophers were deeply shaken by the implication of the Theorem on the interpretations of democratic decisions. A famous political scientist William Riker asserted the meaninglessness of democratic decisions and advanced, unsuccessfully, a liberalism direction for democracy.
After Riker, the new fad among political scientists and political philosophers had been deliberative democracy, popularized by Rawls and Joshua Cohen in the US and Habermas in Europe. Deliberative democracy tries to rescue meaning (epistemicity) in democratic decisions by focusing on deliberation and public discourse. They all believe that the deliberative nature will allow people to consent to democratic decisions. 
Most recently David Estlund combined the epistemic element and consent element together to propose a normative consent theory of  democracy. To Estlund, democracy is the only legitimate authority as it satisfies two necessary requirements for political authority (the moral right to obeisance): decision is wise, and people are treated as equals.    

