The other two answers are great, but I wanted to add another perspective on the reasoning for why.
So the short part of the answer is that, no, a Fighter cannot use any actions while Surprised.
Salteris' answer explains that you can activate Action Surge, but you can't use actions while being surprised, and since Action Surge gives you exactly that, you can't use it
Nitsua60's answer explains that because the rules for being Surprised state that you specifically cannot take any actions, you are not allowed to activate your Action Surge to get an additional action because you can't use it in the first place.
The way I read those, Action Surge is an ability that you "activate" when you want to take another action. I am writing here to offer a different perspective. Let us look again at the Action Surge text in whole:
Action Surge
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action. †
Once you use this feature, you must finish a short or long rest before you can use it again. Starting at 17th level, you can use it twice before a rest, but only once on the same turn.
So it definitely says we have the ability to take an extra action on our turn, which we all understand. What I think we can look at differently is how we treat this extra action. The other answers seem to look at Action Surge like a class feature that you must activate, but it's starting to look to me like the Action Surge Fighter class feature gives you a resource, and defines a limit on how much of that resource you have. At 2nd level, you have one extra action you can use on your turn, and once you use that you won't get another one back until you rest. At 17th level, you get a second one, and you can still only use one per turn as long as you have one.
This could then be treated similar to a Druid's Wildshape, a Paladin's and a Cleric's Channel Divinity, or a Barbarian's Rage. Each of those features describes a thing you can do, what it might cost you (if anything), and how many times you can use that feature. Their limits are defined with different rules, but they still follow a similar structure. Barbarians get n rages between long rests as defined by their Barbarian levels on the Barbarian Class Table. Clerics and Paladins have a set number of Channel Divinity uses which changes so infrequently that it describes the levels that it changes. Druid's Wildshape can be used only twice between rests, and that doesn't change until 20th level.
But my point is, those can be considered limited resources, not just features you "activate". I suspect Action Surge works in a like manner, as a resource you expend, not exactly an ability that you activate when you want to use it. Another point, if minor, that might provide some evidence for this perspective is the lack of any conditions for taking the additional action.
Moving on
Back to the main point here. A level 2+ Fighter has the feature Action Surge, which provides the Fighter an additional action they can use on their turn once between rests. Surprise says that if you are surprised, you can't move or take an action (and by extension a bonus action) on your first turn of combat. You normally have an Action and an amount of Movement available to you, and Surprise says you can't use them. The Fighter having multiple actions doesn't change the fact that you can't use any of them.
I will apologize here for making yet another wall of text. But I do hope that it adequately explained this alternate viewpoint that we can take.
† - This is an unrecorded errata found in recent printings of the PHB. Your copy of the book may include the following: "On your turn, you can take one additional action on top of your regular action and a possible bonus action."
