No, Sauron couldn't simply reach out and take the ring.
Nowhere in Tolkien's Legendarium is it ever stated that the Ainur could simply reach out and take things from a distance. Further, Sauron used the Ithil-Stone to gaze out in search of the ring, which required great will power and could not been done constantly, it would not allow him to take physical possession of the ring.
I think the quote to answer your question is the one you provided (emphasis mine).
He was kneeling in clear sunlight before the high seat.
A black shadow seemed to pass like an arm above him; it missed Amon Hen and groped out west, and faded.
Then all the sky was clean and blue and birds sang in every tree.
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
Frodo's location, upon the seat of seeing on Amon Hen not only allowed him to gaze far, and in his gaze spot Barad-Dûr
The world seemed to have shrunk and fallen silent. He was sitting upon the Seat of Seeing, on Amon Hen, the Hill of the Eye of the Men of Númenor...
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
The text preceding your quote further suggests that Sauron was searching for the Ring, and wasn't able to physically grab it
Then at last his gaze was held: wall upon wall, battlement upon battlement, black, immeasurably strong, mountain of iron, gate of steel, tower of Adamant, he saw it: Barad-Dûr, Fortress of Sauron. All hope left him.
And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir – he threw himself from the seat, crouching, covering his head with his grey hood.
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
This passage suggests that although Sauron was searching for him,
"He knew it had become aware... It leaped towards him... he felt it"
This snippet discusses Sauron's search for the ring, with Sauron become aware of Frodo's gaze (implying he sees someone searching) and that Sauron's search "leaped towards him", and Frodo could feel his stare.
This leads me to conclude that he would've been unable to grab the ring directly from Frodo from afar, and was only able to pinpoint Frodo's location. To, in turn, send the Nazgûl in pursuit of the ring. In addition, the "Shadow" was, IMO, a literary representation for Sauron's gaze, and possibly a portrayal of his evil (as Blackwood's answer suggests)