On what basis should Christians oppose child marriage? In most of the western countries the minimum age for marriage is 18 years. (Some would like to argue that this could be 16 or 15 but none would argue it to be 9)

As a Christian I take pride in the fact that Christianity does not support the horrible custom of child marriage (which is still practised in some parts of the world). 
It is one of those things that you don’t think much about because it seems so very basic. 

However it recently crossed my mind that I’ve never seen a Biblical backing to it. Could it be that it’s only a cultural concept that I think of as a Christian concept?

So my question: Does the Bible define an age of marriability? If not, then on what Biblical basis should Christians oppose child marriages?
Indeed, AFAIK the Bible doesn't make any overt statements on this subject. It is, however, usually accepted that Mary was around 14 when giving birth, and was married at that time. In many ways it is perhaps fortunate that the Bible doesn't wax lyrical on the subject; your description of child marriage as a "horrible custom" sounds largely a product of culture (or dare I even say, "moral relativism"?). There is obviously also an issue of comparison, with radically different life expectancy and life-style (education in particular, and a harsher life, leading to a shorter childhood), historically at an age of around 14 (not the younger 9 that you cite) the majority of females would be fertile (or rapidly approaching fertility), about as educated as they were ever going to get, and with limited years ahead.

Islam, of course, has a glitch here in that it needs to accommodate the marriage of Aisha (often taken to be around 9) with the view of Muhammad as the ideal Muslim, which makes it rather hard for Islam to make a statement that such marriage is inherently wrong. However, even in Christianity (and OT in particular), the fact is that many (not all) of the laws around marriage are more about property laws (with the wife being the property) than of human rights-of / decency-towards the wife.

This leads me to conclude that we aren't going to get much direct and literal guidance from the Bible on this subject, and must resort to other arguments. Perhaps arguments such as these, or from unicef.