The Door of Time cannot be closed from the inside.
How do you know?
if he returns the sword, he should go to the past, and then he wouldn't be sent by Zelda, but by the Sword
If the "door between times is closed", returning the sword should have no effect.
And we know the way of time traveling using Zelda's powers and the Master Sword is different: while with the second, Link doesn't have the ToC, with the first he does.
No, actually, we don't know whether Link would have had the ToC in the past with the Master Sword method. We don't see any direct differences in the time travel itself except for the apparent point in time that is Link's destination--presumably before
he pulls the Master Sword. He still grows younger as he goes back in time, he still appears in the Temple of Time, a column of blue light still heralds his coming.
This is why I prefer to leave the time travel mechanics alone--we expect that the time travel should be different, but, when he arrives back in the Temple of Time, we don't see any visible differences, aside from the fact that he's not got his hands on the sword (which, again, is more likely due to the fact that he's not just been sent to the last point he touched it than anything else).
So either a split timeline or a single timeline can
proceed from OoT, neither of which is directly or exclusively evidenced from anything we see in game, since what we see in-game is either:
1) so limited that we don't get a complete picture of what has happened;
2) so similar to what we've seen throughout the rest of the game that it is difficult, nigh, impossible, to tell whether there's a notable difference in the time travel at the end, other than the point of destination
They're both possible, both just as likely, but cannot both be intended by the creators. And here is where I take my leave from the single-split argument once again.