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Sauron made the One Ring to rule them all by transferring some of his essence into it as a power amplifier.
Morgoth also did this in a different way, using the whole of Arda as a Ring.
Why didn't the Valar do the same, in order to gain power to resist Morgoth and Sauron?
Because Eru Iluvatar forbade it.
In the Ainulindale, "The Music of the Ainur" Tolkien writes that Iluvatar directed to Ainur to make a great music and they did, but Melkor wanted to do things his own way and started his own music in competition
and straightway discord arose about him, and many that sang nigh him grew despondent, and their thought was disturbed and their music faltered; but some began to attune their music to his rather than to the thought which they had at first. Then the discord of Melkor spread ever wider, and the melodies which had been heard before foundered in a sea of turbulent sound.
This is Melkor/Morgoth's first rebellion and, importantly, it is a selfish rebellion against Iluvatar and a domination of others' wills. As this first, so all the later.
The Valar were also tempted to dominate others' wills -- with the best of intentions of course! -- but this never went well:
Elves and Men are the Children of Iluvatar; and since they understood not fully that theme by which the Children entered into the Music, none of the Ainur dared to add anything to their fashion. For which reason the Valar are to these kindreds rather their elders and their chieftains than their masters; and if ever in their dealings with Elves and Men the Ainur have endeavoured to force them when they would not be guided, seldom has this turned to good, howsoever good the intent.
This is as close as you can get to the moral basis of Arda: Even "the Valar are to these kindreds rather their elders and their chieftains than their masters". The Great exist and are genuinely great, but this does not give them the right to dominate others.
The rebellion of Morgoth and Sauron was basically against this moral principle. Morgoth, out of a lust for personal power, and Sauron (initially at least) out of a techie's dream of a perfect, tidy, efficient world. They had their dreams and were the sort of beings who would follow those dreams no matter the cost to others. Both Morgoth and Sauron chose the path of domination.
How? Well, initially armies and the use of their own power -- the Silmarillion is full of those stories -- but Morgoth and then Sauron discovered that they could pour their essence (spirit? being? strength?) into a material object to give themselves much greater mastery over the world.
Morgoth, who was a "bitter cold immoderate" sort of guy who liked the "turmoils of the heat and the cold that had come to pass through him" wanted to dominate reality and poured his strength into Arda itself and gained power over the physical world. Sauron, who initially just wanted people to work together to build a better world (under his direction, of course) eventually poured himself into the One Ring so that with it he could "encourage" people to work together under his direction. I.e., so he could dominate the wills of others.
It is said that Sauron originally meant well, but because he chose domination as a tool, his own self became corrupted and in some respects the Second Age is the chronicle of Sauron's final corruption. Note that at the time he forged the Ring he could still appear fair and persuade people. After making and using this tool of domination to rule Middle-earth and to destroy Numenor, he permanently diminished himself and could only appear as a Dark Lord.
The Valar knew that domination -- the use of power to force Men and Elves -- had been forbidden by Iluvatar. And if they were tempted to ignore that they had what happened to Morgoth and Sauron when they forged their Rings to see.
Why would they even consider making their own?
Sauron's purpose with the Rings of Power was to dominate the Elves.
From The Silmarillion, in Of the Rings of Power and the Third Age:
Men he found the easiest to sway of all the peoples of the Earth; but long he sought to persuade the Elves to his service, for he knew that the Firstborn had the greater power; and he went far and wide among them, and his hue was still that of one both fair and wise.
...
In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power. But Sauron guided their labours, and he was aware of all that they did; for his desire was to set a bond upon the Elves and to bring them under his vigilance.
The Valar had no need nor desire to dominate the Elves in that way, so they didn't need a One Ring equivalent.
To directly answer the body of the question, "Why didn't the Valar do the same, in order to gain power to resist Morgoth and Sauron?" the Valar were capable of resisting Morgoth and Sauron without amplifiers. The problem was when they did so directly, in The War of Wrath, much of the land was wrecked and sunk. They didn't lack focused power.
I have reviewed the other answers, and find something lacking: what could something like a magic ring do? The One Ring was clearly designed to rule the world, and specifically the elves. I have no hint of what the Nine Rings did, and the Seven apparently increased greed? But the Three, which could ward off the decays of time and postpone the weariness of the world weren't ever touched by Sauron:
Therefore the Three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.
They used his knowledge, but not his aid, in making the Rings. And when Sauron was vanquished and the Ring cut from him, then the elves started using the Rings again. And to good purpose! But their power was not to conquer, but preservation. One can't tell what wonders such techniques might produce that didn't contribute to war, but one can imagine things like physical and mental healing, strengthening of will, increasing happiness, etc.
Of course, the One Ring was made in such a way that anything else made with the same principles was linked to it:
but secretly Sauron made One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last.
So the elves or the Valar could have made other such objects to do many useful things, and all would be useless when the One Ring was destroyed.
It is also possible that Sauron was the only one that figured out how to make such items, and never shared that knowledge with the Powers in the West. He shared some of it with the elves, so they could make some on their own. But Celebrimbor, the maker of the Three, was slain soon after Sauron put on the One, and to whom did he teach anything in that time?
In conclusion, the one who knew the most about making these things was Sauron, not the Valar, and it is likely no one else was able to figure out enough to make useful items like them.
