Why did monarchy become common practice after the fall of Rome?

1) Long before Rome fell it had abandoned Republicanism. After Diocletian and the Crisis of the Third Century the Emperors no longer felt any need to consult with the Senate. The Senate's role in government and in the legitimacy of the Empire was symbolic. This is the first flaw in the question - the transition between Republic and Monarchy occurred during the Roman Empire.

2) Greek Democracy fell to the Macedonians, although a more full discussion would include that (a) the Greeks abandoned democracy themselves more than once, and (b) Greek Democracy was actually an oligarchy - voting rights were invested in a small minority of the inhabitants.

3) Europe didn't pursue absolute monarchy for centuries. In the comments many others have noted that the governance model from the fall of Rome to the rise of absolute monarchy explored a variety of models. Most common, as I recall, was for those who wielded effective military power to select one of their number to exercise leadership, but that selection was limited and revocable.

4) Ultimately Monarchy wins out for a couple of reasons. Democracy and Republics suck at war. War requires effective command, and Democracy tends to provide a diffuse and inchoate command structure - Rome (and if I recall Greece) delegated to a dictator during a war. (I think one can argue that Democracy sucks at foreign policy in both peace and war.) Monarchy provides advantages for the organization and governance of societies of a specific size and technological level. hat tip to @bilbo_pingouin who suggested this revision Monarchy provides for a (relatively) stable succession, coherent governance, and strong effective states. Hat tip to @TheHonRose This changes somewhat in the modern age; @TheHonRose correctly points out that Churchill defeated Hitler; that democracies defeated empires in WWII. But OP is asking about a pre-modern time where war was fought & won with weapons is different from a modern time where war decided by industrial production. There is a fascinating essay there but (a) I'm not truly qualified to write it, and (b) it is non-responsive to OP's question. I'll stand with my thesis that during the pre-modern age, working with economies based on agriculture and land, that monarchy (and vassalage) is a more effective command and control institution. Rome was an aberration - Rome developed an educated class broad enough and deep enough to take advantage of a form of democracy, and an economic infrastructure that surpassed simple agriculture. (another interesting essay, but probably off topic).

Update - It occurs to me that I've neglected to address the underpinning of OP's question. Why monarchy over democracy? Fukyama's Origins of Political Order argues that democracy requires some relatively sophisticated institutions. (I'm abridging a book into a sentence; I'm aware of the injustice I do to his argument). Democracy is built on the notion that the citizens trust the state. In an era of poor communications, and unbelievable wealth gaps, it is difficult to assemble any meaningful demos. (Stop whining about today's 1%'ers; imagine life in Charlemagne's time, when 99% of the population legally could not travel from their home and would almost never speak to anyone outside their home village.) Greek democracy worked because it was a tiny oligarchy of citizens who knew one another. The same was true for the Roman Senate - a small coterie of families who knew one another, supported by tribes (who claimed kinship). Rome couldn't maintain the cohesion and degenerated to an absolutist monarchy, followed by a military dictatorship. There was no hope to build the kinds of institution and societal trust mechanisms that would enable democracy in societies with that kind of weak communications and cooperation.