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Yes that is true. However, their decision to march upon the Black Gate was based mainly on their own speculation. They didn't know what Sauron would do next after his defeat on the fields of Pelennor. Yet they sought to help Frodo and Sam in any way possible; wanted to draw Sauron's eye away from his own land. (Although they didn't even know if Frodo and Sam still lived)

At the point in the book when Aragorn and his forces stood before the Black Gates, Frodo and Sam where walking up Sauron's road to Sammath Naur on the morning of March 24th or 25th.

Looking at the dates in the Appendix of the LOTR as well as the dates written in the Atlas of Middle-earth it happens that Aragorn and his forces assail the Black Gates on the very day when Frodo is most visible and closest to Baradur. If Aragorn hadn't shown up any later The Great Eye would have spied Frodo and Sam eventually. The Ring's power and influence grew stronger as it neared Orodruin.

Say Aragorn refused Gandalf's advice before they even set out from Minas Tirith...I think Sauron would have taken the Ring Frodo and tortured him as he promised; "He was dear to you, I see. Or else his errand was one that you did not wish to fail? It has. And now he shall endure the slow torment of years, as long and slow as our arts in the Great Tower can contrive, and never be released, unless maybe when he is changed and broken, so that he may come to you, and you shall see what you have done." -The Mouth of Sauron from the The Black Gate Opens.

Say Aragorn accepted Sauron's terms then he would have delivered all the free lands, whether his or not, unto the whims of Sauron; and all Sauron desired were thralls. The apocalypse of Middle-earth would be an interesting thread to consider.

I think Sauron's gaze was upon Minas Tirith during the battle The Seige of Gondor and he indeed waited after he was informed of the fall of his Captain, The Witchking of Angmar. After he learned that his forces were beaten by Gondor he waited with anticipation for the Lords of the West's next move. The Dark Lord learned that they marched for the Black Gate on March 18th and were finally assaulted after many days of uneventful travel on March 21st: "Ever and anon Gandalf let blow the trumpets, and the heralds would cry: 'The Lords of Gondor are come! Let all leave this land or yield them up!' But Imrahil said: 'Say not The Lords of Gondor. Say The King Elessar'. But none answered the challenge.
Nonetheless, though they marched in seeming peace, the hearts of all the army, from the highest to the lowest, were downcast, and with every mile that they went north foreboding of evil grew heavier on them. It was near the end of the second day of their march from the Cross-roads that they first met any offer of battle. For a strong force of Orcs and Easterlings attempted to take their leading companies in an ambush; and that was in the very place where Faramir had waylaid the men of Harad, and the road went in a deep cutting through an out-thrust of the eastward hills. But the Captains of the West were well warned by their scouts, skilled men from Henneth Annun led by Mablung; and so the ambush was itself trapped. For horsemen went wide about the westward and came up on the flank of the enemy and from behind, and they were destroyed or driven east into the hills." (The Black Gate Opens, LOTR book V, chapter X) These that were driven into the eastern hills were the ones that alerted Sauron that the Lords of the West were indeed on their way to treat with him and this drew the Dark Lord's gaze in earnest to the West while Sam and Frodo crept almost just beneath his gaze from the North.

    

 

    Sauron's desire was to destroy the West and the Heir of Isildur, (remember Aragorn showed himself and his reforged sword Narsil-into-Anduril) so Sauron delighted in his coming with such little force and Gandalf's words proved true, knowing the mind of Sauron well: "'Surely,' he (Imrahil) cried, 'this is the greatest jest in all the history of Gondor: that we should ride with seven thousands, scarce as many as the vanguard of its army in the days of its power, to assail the mountains and the impenetrable gate of the Black Land! So might a child threaten a mail-clad knight with a bow of string and green willow! If the Dark Lord knows so much as you say, Mithrandir, will he not rather smile than fear, and with his little finger crush us like a fly that tries to sting him?'
'No, he will try to trap the fly and take the sting,' said Gandalf. 'And there are names among us that are worth more than a thousand mail-clad knights apiece. No, he will not smile.'
'Neither shall we,' said Aragorn. (The Last Debate, LOTR book V, chapter IX).

In other words: I think Sauron was only as good as his information and I also think that his far reaching gaze was due in large part to his own perceptions coupled with his usage of the Palantir for it was also said that his gaze can pierce cloud, stone and flesh which was the power of the greater Palantiri.
And remember Huan: He had the tokens of the spies and nothing more; The Mithril shirt and the Elvish blade but no spies so he could not account for Sam and Frodo but decided to put doubt in Aragorn's heart because he assumed Aragorn possessed the one Ring. All else could wait...all his thought was bent on it.

It does seem that Sauron did not even contemplate the idea that the Ring would be brought in stealth to Mt Doom. If that idea had ever occured to him he would have surely secured Mt Doom in some way, perhaps with patrols or even by sealing up the Sammath Naur. However it does seem likely that Sauron would have spotted them in that one instance had it not been for the distraction provided by Aragorn and co.

I don't think I agree with the last post. No, Sauron was not like Morgoth.
From the Myths Transformed (Morgoth's Ring):

"...when Melkor was confronted by the existence of other inhabitants of Arda, with other wills and intelligences, he was enraged by the mere fact of their existence...His sole ultimate object was their destruction". Concerning Sauron, however, "...it was creatures of earth, in their minds and wills, that he desired to dominate [as opposed to destroy]". And then, "He did not object to the existence of the world, so long as he could do what he liked with it. He still had the relics of positive purposes, that descended from the good of the nature in which he began: it has been his virtue (and therefore also the cause of his fall, and of his relapse) that he loved order and coordination, and disliked all confusion and wasteful friction".
Even more interesting, "...though the only real good in, or rational motive for, all this ordering and planning and organization was the good of all inhabitants of Arda (even admitting Sauron's right to be their supreme lord), his 'plans', the idea coming from his own isolated mind, became the sole object of his will, and an end, the End, in itself".

I can imagine that he would not have destroyed Gondorians had they submitted to his terms. At least there is a parallel with the Numenoreans: "But though Sauron's whole true motive was the destruction of the Numenoreans, this was a particular matter of revenge upon Ar-Pharazon, for humiliation. Sauron (unlike Morgoth) would have been content for the Numenoreans to exist, as his own subjects, and indeed he used a great many of them that he corrupted to his allegiance".