There are a number of issues to consider and I will touch on some of them. This answer is not complete.

First I want to point out that no General wanted to follow a pure Mediterranean strategy. This accusation was often hurled at the Brits, but its not true. The British CIGS was always explicit about the goals of the Mediterranean Strategy (as early as 1940), they are as follows:

1. Open the Mediterranean for shipping

2. Draw German reserves South

3. Knock Italy out of the war

4. Hopefully bring Turkey into the War on the allied side

The eventual invasion of Germany was to happen through landing in Northern Europe.

The Americans were of course far less interested in this strategy and on all the conferences it was a battle between how much resources would be put into this strategy.

So nobody had a real plan to end the war without a invasion of Northern Europe.

Now lets address why exactly. One of the major reasons why the Brits wanted to draw German troops South is because how hard it is to supply them there, and the move them, if needed, to another front. Moving troops in on an East-West axis in Europe is easy, because of the extensive railway lines.

Moving troops on a South-North axis is considerably harder. You have the Alps smack down in the middle of Europe, posing a considerable barrier and in the middle of the Alps you have a neutral country, Switzerland, that you could not move your troops through. Once you are over the alps there is plain, but shortly after that, if you go South in Italy, or the Balkans, you will have mountains terrain again.

So for the same reason British Strategist wanted to draw German troops South, it would have been unviable to do the full invasion of Europe from there. Consider this map:

https://upload.wikimedia.org/wikipedia/commons/d/d9/Alpenrelief_01.jpg

Clearly its not possible to directly go North from Italy. So you have to go around them, most likely into the South of France. Once you are there, you have to fight your way up France, and then you are at the same point that you would have been, if you had just invaded Europe in the North, just that your lines of supply are far longer, more complex, you would have to do it without as much air support.

There were some suggestion to push on to Vienna (most notably from Churchill) once they had reached the North of Italy, but by then they had established themselves in France and the Strategy called far reducing the effort from the South. The troupes that the Germans had dispatched South, were now bound there and could not help in the defence of Germany proper. The Strategic imperative of the Offensive was successful and further pushing would be wasting resources on strategically unimportant goal (of capturing Vienna).

tl;dr: From a strategic perspective its much easier to invade from the plains of Northern Europe, its also better because of logistics,terrain and air support.

Sources:

• Alanbrooke War Diaries 1939-1945: Field Marshall Lord Alanbrooke

• Allied Master Strategists: The Combined Chiefs of Staff in World War II

• The Path to Victory: The Mediterranean Theater in World War II

As a supplement to /upanick21, I wanted to suggest that even if the Allies had decided by the Summer of 1943 to pursue a Mediterranean strategy first, the events in Italy made that plan sour. Progress up the peninsula was very slow going, and by January of 1944 the attempt to open up a beachhead at Anzio was not going very well at all. Invasion of northern Europe must have been a very difficult decision given the events at Anzio. The German resistance in Italy was supposed to be inferior to that in France, and it was thought that the allies had hit Anzio with complete surprise.