There's no reason to suspect that he can, officially, override the Hogwarts Board of Governors. For all intents and purposes, Hogwarts seems to function as a private school - it seems that it must meet the requirements of the Ministry but there's no reason to suspect the Ministry of Magical Britain has any more say over Hogwarts hiring practices and internal operations than Congress or the President have over a private school in the US.
That's not to say the Minister (or even the Ministry) is incapable of doing such a thing - in later books, they legislate specific powers over the school. That said, those powers they give themselves are given by legislation that (presumably) must pass some sort of overview or be voted upon. The times we see it mandated are times when the government is (and is supposed to be seen as) overstepping its bounds.
It's certainly possible that the Minister of Magic, as an important political figure who (in theory) has the support of a majority of the population, could influence such things. Important politicians tend to be wealthy and have powerful friends, many of whom owe them favors or who would like favors owed. In this case, however, Lucious Malfoy is similarly powerful and capable of exerting pressure, has a more strongly vested interest in a particular outcome, and the Minister (Fudge) is both under his sway and extremely weak.
So, to sum up: NO, the Ministry can't really make such determinations, until they give themselves that power (likely illegally) and Fudge is incapable of standing up to the will of Malfoy to make it happen anyway.