No. Social and economic intervention by the modern state began long before universal suffrage became widespread. For example, in Britain, "collectivist" measures were introduced when property and sexual restrictions on voting rights still existed. The centralist and hierarchical nature of "representative" democracy means that the population at large has little real control over politicians, who are far more influenced by big business, business lobby groups, and the state bureaucracy. This means that truly popular and democratic pressures are limited within the capitalist state and the interests of elites are far more decisive in explaining state actions.
The "New Deal" and the post-war Keynesianism measures of limited state intervention to stimulate economic recovery from the Depression were motivated by more material reasons than democracy. Thus Takis Fotopoules argues that "[t]he fact . . .that 'business confidence' was at its lowest could go a long way in explaining the much more tolerant attitude of those controlling production towards measures encroaching on their economic power and profits. In fact, it was only when — and as long as — state interventionism had the approval of those actually controlling production that it was successful" ["The Nation-state and the Market", p. 55, Society and Nature, Vol. 3, pp. 44-45]
An example of this principle can be seen in the 1934 Wagner Act in the USA, which gave US labour its first and last political victory. The act made it legal for unions to organise, but this placed labour struggles within the boundaries of legal procedures and so meant that they could be more easily controlled. In addition, this concession was a form of appeasement whose effect was to make those involved in union actions less likely to start questioning the fundamental bases of the capitalist system. Once the fear of a militant labour movement had passed, the Wagner Act was undermined and made powerless by new laws, laws which made illegal the tactics which forced the politicians to pass the Wagner Act in the first place and increased the powers of bosses over workers.
Needless to say, the implication of classical liberal ideology that popular democracy is a threat to capitalism is the root of the fallacy that democracy leads to state intervention. The notion that by limiting the franchise the rich will make laws which benefit all says more about the classical liberals' touching faith in the altruism of the rich than it does about their understanding of human nature or their grasp of history. The fact that they can join with John Locke and claim with a straight face that all must abide by the rules that only the few make also says a lot about their concept of "freedom."
Of course some of the more modern classical liberals (for example, right-wing libertarians) advocate a "democratic state" which cannot intervene in economic matters. This is no solution, however, as it only gets rid of the statist response to real and pressing social problems caused by capitalism without supplying anything better in its place.
Anarchists agree that the state, due to its centralisation and bureaucracy, crushes the spontaneous nature of society and is a handicap to social progress and evolution. However, leaving the market alone to work its course fallaciously assumes that people will happily sit back and let market forces rip apart their communities and environment. Getting rid of state intervention without getting rid of capitalism and creating a free, communal society would mean that the need for social self-protection would still exist but that there would be even less means of achieving it than now. The results of such a policy, as history shows, would be a catastrophe for the working class (and the environment, we must add) and beneficial only for the elite (as intended, of course).
The implication of the false premise that democracy leads to state intervention is that the state exists for the benefit of the majority, which uses the state to exploit the rich minority! Amazingly, many capitalist apologists accept this as a valid inference from their premise, even though it's obviously a reductio ad absurdum of that premise as well as going against the facts of history.