No.
Castling Rules
I think the bit that hasn't been explained sufficiently is that the king cannot move through check while castling.
With the pawn on e2, it is effectively 'attacking' the f1 square meaning that the king would have to move through check in order to complete the castle and so it is not allowed.
One point I had in mind is that it can be argued that a pawn only "attacks" or can move to, a square diagonally in front of it, which is occupied by an enemy piece. In my original scenario, the relevant squares are not so occupied. If this is so, surely the King is not moving through check?

If the King was on f1 he would be in check by the pawn.
By castling he would be moving through that square and hence it would be an illegal move.
My advice would be to take the pawn with the king

Yes the king must move through the threatened f1 square. The king can never move to or through a threatened square. This rule in casting is to keep the principle of the king not moving to an attacked square at all times, even when it moves outside its normal movements I.e. casting, when it moves two and not one square. This is also why a rook CAN do that, because it's normal movements allow for it too.
In the diagram, white can castle long (queenside/O-O-O) but cannot on the kingside. Even though the king is moving two squares, it is moving through the intermediate square first and if he would be in check at that point, it is a no go.
Another similar situation is during the decision whether to take en passant or not. Even though the the pawn has moved two squares, it behaves as if it is moving through one square and then the next and so it can be captured.

I like the coincidence of the two words casting in post 11 and the baby on the picture who can be casting the pieces !

Yes the king must move through the threatened f1 square. The king can never move to or through a threatened square. This rule in casting is to keep the principle of the king not moving to an attacked square at all times, even when it moves outside its normal movements I.e. casting, when it moves two and not one square. This is also why a rook CAN do that, because it's normal movements allow for it too.
When you put it like that, it makes a very good explanation for the en passant rule too. The difference is that a pawn is legally allowed to move to an attacked square, but if excercising its rights to move two squares instead of one (outside its normal movements) it can, and should, be subject to the same rules as if it had moved one square. It kind of makes you wonder why pieces can't capture pawn en passant.
To return to this vexed subject again, my point is that a pawn can only move to the squares diagonally to the left or right in front of it, if the square involved is occupied by an opposing piece, ie an opposing piece is there to be taken. In my original position, this is not the case. My contention, therefore, is that the King would not be moving through a threatened square, or moving through check.

To return to this vexed subject again, my point is that a pawn can only move to the squares diagonally to the left or right in front of it, if the square involved is occupied by an opposing piece, ie an opposing piece is there to be taken. In my original position, this is not the case. My contention, therefore, is that the King would not be moving through a threatened square, or moving through check.
My point is that a bishop can only make a capture on a square that is occupied by an opposing piece, i.e. an opposing piece is there to be taken. In my position, this is not the case. My contention, therefore, is that the King would not be moving through a threatened square, or moving through check.

So, actually, your contention is that pawns don't threaten unoccupied squares while every other piece does.
Which is odd on its own, but I think you'll also struggle to reconcile this with en passant.
To visualize the rule differently, pretend whn castling the King gets two moves - and moves one square at a time, not two squares at once - stopping at f1 while he takes a breath. Can't do that, the pawn would take the King.
But he doesn't get two moves - he moves two squares and then the rook jumps over him.

Actually he asks it in the OP, he just restates it in #8. I think a lot of people didn't catch that actually :p
I didn't like the "imagine the king is walking" stuff because that's too weird and abstract. My imagination (should) have nothing to do with the rules. I liked your answer in #13 because I think if you use his argument in an inane way, he's more likely to understand.
And yes, I saw immediately this was over a year old :p I laughed at the "slow learner" comment.
Dear All,
I hope this is not regarded as a silly question. The scenario (improbable, I agree) is that white's King is on its original square of e1 and the kingside rook is on h1 and neither has moved. A black pawn is on e2. There are no white pieces between the white King and above rook, no other black pawns on the second rank and no other black pieces attack or control any of the squares e1, f1, g1, or h1.
Bearing in mind that a pawn can only reach a square diagonally in front of it to left or right if said square is occupied by an opposing piece, can white castle on this side?