No, as democracy can be combined with every economic system. It is an error to think, democracy and free markets are impossible without each other. Democracy can be combined with every way of economy, as free markets can be combined with undemocratic systems.
That said: I'm not even sure economic quality is impossible with free markets, although it is pretty hard to achieve.

Complete economic equality is an impossible goal anyway under any system. Someone will always have more than the others. The thing is now the current inequality is just far too high as average well wages have stagnated and the rich just seem to be paying themselves over the top bonuses which go straight into an off-shore bank account doing nothing.

I personally want a social democracy to at least make the playing field a bit fairer. This includes a decent welfare state, as well as free and accessible education and healthcare. There must be a mixed economy with regulations.

Unless people think everyone has the same amount of money, the answer is obvious.
In a democracy, if you profit, it means someone else in the world has lost out.

Capitalism is the best thing ever conceived by man. Equality only exists in mathematics.

No. Though what do you mean by economic equality? There can never be economic equality when society values some things people do more than others. A doctor will always be valued and rewarded more than the person who cleans his/her office, and democracy has nothing to do with that.

If you actually mean an economy that is fair and where the gap between rich and poor is much narrower than what we see today in pretty much every country, I think this is only possible in a democracy. But not a democracy of the sort we have operating in the world today. What we have in today's democracies is too much money talking, and politicians only thinking far enough ahead to get themselves re-elected. That is not a recipe for a fairer economy and a more equitable distribution of wealth.

Democracy exactly leads to economic inequality. WHy? BEcause we elect people to govern in our best interest. Putting power in the hands of the minority gives rise to inequality.
It's for the same reason the communism doesn't work.

As long as the minority have the power, the minority will have the wealth. It's simple.

In order for there to be economic equality, every decision would have to be voted for by the people themselves.

PS: oh and to answer your last sentence, yes, the check and balance is supposed to be there to allow us to elect someone else, but in truth, the way it is set up, it only takes one bad election to fuck up everything irreparably, and the are measures in place that prevent future leaders from undoing policies.

I don't think economic equality is achievable, period, in a capitalistic society. The largest thing I took from my senior Econ class (don't make fun of it! I did go on to go to Purdue) was that in order for there to be rich people, there must be poor people.

I've heard conservative pundits say "see, they think that if I make $100, that's $100 they can't get. But in actuality, we want to increase the amount of money for everyone". But that can't ever happen. Money is finite. There has to be, and will be, a difference in income in people. Because consumers control prices. Even if we all made the same amount of money, if I want an item more than the next guy, I'll pay more, meaning the supplier will have more money than he "should". That simple concept will eventually make a gap in income.

Not to mention that the moment everyone starts making even remotely similar wages across the board, there will be screams of socialism, and people will fall back on "why would I go through years of training to become a doctor when I can just work at McDonald's?"

Democracy and economic equality are compatible as long as you use the right regulations'which never happens and never wikl happen.
1st ) you need a gap for maximum wealth & maximum income.
You have to limit the wealth of someone to a max of eg. 50mio dollars and to limit the size of a company(eg. 3000workers or 1billion dollar prod.capacity per years max.As soon as this point is reached you have to split the company into two new units competing with each other)
Than you have to stuck the earnings of the top of the company and the bottom. =no boss should be allowed to earn more than eg. 30+ more than the guy with the lowest income.This will improve insome equality+motivate the boss.

as result you will get several smaller companies+more jobs+better quality+far more millionaires,as there are no more billionaires and this will make lobbyism almost impossible as there will be thousands of companies of a mid-size but no more behemoths.

most problem in democracies are caused by too rich people/too big banks/too big corporations as they were almost always the reasons for non-religious wars.

the main problem with your question is:
there never was a re democracy(though the swiss system is pretty close to it)
there never was economical equality(there can't be equality if you don't start under the same circumstances,but as long sons and daughter get a million legacy from their superrich fathers and as long as wars and fiat money can be printed there never will be)

The problem I have with the answers I'm seeing here is people saying things like "x could happen in a democracy, but we do not have a democracy... to much corruption, blah, blah, blah".

The issue I take is that it assumes that a "pure" democracy, however they define it, could actually, realistically exist. When it can't. Whenever you have a system where you give one group of people consent to trample on the rights of everybody else... whether that power be appointed, elected, or inherited - it's inevitable that people will exploit this power. It's not that they are bad people, per se, it's that just about all humans would react in the same way when put into a position of power.

"Limited" or "Constitutional" Government is bullshit. It requires such a high level of vigilance from the general population that it's realistically an impossibility.

As per my actual answer to your question, well, then, the answer is clearly no. There are two ways you can mean by "equality", equality of outcomes, and equality of opportunity.

Equality of outcomes doesn't seem like it'd be something that genuine, intelligent, people would actually want. I can't fathom why anybody thinks this would be a good result. As a previous poster mentioned, why would you want the doctor to have the same outcome as the guy who cleans his desk? It's madness. However, if it is something that you do want (say you've been hit on the head with a hammer, or went through state education...):

Still impossible, on the very face of it. In order to enforce such equality will require a state so large that you'll HAVE to create a political class, who will have those powers of the state at their disposal. As a result, the political class will be the wealthy elite.

As for equality of opportunity. This is something that I think more people can get behind, but... still incompatible with democratic systems. Democratic systems will lead to corruption, which will lead to some people striving to get their "leg up" in the system. Those in the so-called 1%, so-to-speak, will be able to rig the system to their advantage. Getting all kinds of regulations and taxes and laws passed which make it much harder for new entrants.

As the size of the state inevitably grows, it'll seize control of the monetary system (Fed in US, BoE in UK, etc), and ultimately, use that to push more wealth into the hands of the friends of Government, at the expense of rapidly increasing prices for the lower classes.

They'll throw some scraps at those guys, in the guise of welfare and tax breaks, to keep you happily on the farm, but it's a mere fraction of what they took from you in the first place.

This is false. Just about everything you said.

In a purely capitalistic system, there's only one way to make money: by trading something you have for it. How can there be losers in trade? If I thought I was going to get poorer through by trading something, I would never do it! When I go into a shop and buy X with $Y, I'm not $Y poorer, I'm X - $Y richer.

Value is subjective, everybody perceives things to be valued different things. This has to be true, or trade simply could not exist. When I buy a burger in McDonald's, it's because I value the burger more  than the money, and when McDonald's sells me the burger, it's because they value the money more than the burger.

Everybody wins in trade, everybody gets richer, it's a win-win situation. There are no losers.