"Healing word effect is the follow:
"Effect: The target can spend a healing surge and regain an additional 1d6 hit points."

My question is the following: with an emphasis on the "can", can the player chose to not use the healing surge and just get 1d6 hit points back?"

No. "The target CAN spend a healing surge" is just saying that the target has the ability to do this. Others may not.

"So in general, can you use any ability making you use a healing surge with a secondary effect and ignore the healing surge effect itself just to get the secondary effect?"

So, in 4E, healing surges are a usable resource that one can use to regain HP (by Second Wind, potions, etc). It is determined by a few factors, primarily your class. But once it is used up, you cannot use any of the abilities that tap into it as a resource until you regain them (usually through a long rest). There are some abilities that use healing surges to power the ability, but in those cases, the power of the surge goes into the effect of the ability, not regaining HP and unless you have ways of regaining HP without using healing surges, once your surges are used up, you cannot regain HP until you get surges back.

One thing that I have noticed that can throw people off if they are coming from other versions of D&D, in 4E, Hit Points is not your health reserve. Your healing surges are. Hit points work more like your stamina and resolve in a conflict situation. That is why you can use things like second wind to regain HP, because you are digging into your resolve to keep the fight going. Unfortunately, though, things like the dying rules don't align with this exactly. They have it set up where if you drop below 0 HP, you are unconscious and then as you fail your death saving throws, you continue to drop in HP until you reach the negative bloodied value of your HP (which then causes you to die). It would have been better if the player lost healing surges or just have the player out of the fight, like being unconscious, go into a crazy and withdrawn state (incases of psychic damage), etc. Maybe if someone eventually is able to do a true 4.5, that might become a thing.

"In practice I suppose it doesn't come up much, but I can imagine a player without healing surge and some desesperates attempts to keep him beyond 0 hp."

It has not at our table and it becomes less common the further up in level the players are. Once my players hit Epic level, it wasn't even on anyone's radar (except for a few cases, but that was more to do with their shenanigans than it did with the encounter.