It doesn't prove anything of the sort.
Well, let's put our minds to it, and try to figure out exactly what happened.
To begin,
1. Doubleday documents the movement of the entire garrison, (see his numbers) family members, personal items and all, out to the docks (see his description).
Can you agree with that part of his narrative?
2. Next, he mentions that they encountered and peacefully passed Confederate troops.
Still with me and Mr. Doubleday?
3. Next he describes the loading of 3 schooners and some row boats. Did you see that?
4. Then, after some other details concerning sentry boats, manned with Confederate troops, he says they arrived at Ft. Sumter.
So, in summary, Mr. Doubleday affirms that there was no military action against their movements out of the fort, over to the docks, the embarkation, or the row over to Ft. Sumter. There were no Confederate troops at the gates of Ft. Sumter, nor any prohibiting their entrance.
Now what about that do you find proving that the Union garrison was under attack?