Was Gandalf intent on taking the Witch-King down?

Question

In Lord if the Rings: The Return of the King Denethor implies that Gandalf might be not strong enough to fight Lord of the Nazgûl.

‘Yet now under the Lord of Barad-dûr the most fell of all his captains is already master of your outer walls,’ said Gandalf. ‘King of Angmar long ago, Sorcerer, Ringwraith, Lord of the Nazgûl, a spear of terror in the hand of Sauron, shadow of despair.’

‘Then, Mithrandir, you had a foe to match you,’ said Denethor. ‘For myself, I have long known who is the chief captain of the hosts of the Dark Tower. Is this all that you have returned to say? Or can it be that you have withdrawn because you are overmatched?’

Pippin trembled, fearing that Gandalf would be stung to sudden wrath, but his fear was needless. ‘It might be so,’ Gandalf answered softly. ‘But our trial of strength is not yet come. And if words spoken of old be true, not by the hand of man shall he fall, and hidden from the Wise is the doom that awaits him. However that may be, the Captain of Despair does not press forward, yet. He rules rather according to the wisdom that you have just spoken, from the rear, driving his slaves in madness on before.

Afterwards, when Gandalf decides to save Faramir instead of helping out in the battle. As he learns of Witch King's death and sacrifices made to it, he also says (emphasis mine)

They looked at him, and for a while he was silent. At last he spoke. ‘My friends,’ he said, ‘and all you people of this city and of the Western lands! Things of great sorrow and renown have come to pass. Shall we weep or be glad? Beyond hope the Captain of our foes has been destroyed, and you have heard the echo of his last despair. But he has not gone without woe and bitter loss. And that I might have averted but for the madness of Denethor. So long has the reach of our Enemy become! Alas! but now I perceive how his will was able to enter into the very heart of the City.

According to that last quote, is Gandalf implying here that if not for Denethor, he was going to take Lord of the Nazgûl down?

I'm aware of several questions (like this one) about whether Gandalf had the power to kill the Witch King (as he was not exactly man), but the focus of this question is what Gandalf intent was, not the power he had.

Answer 1

Yes, Gandalf believed he had to face the Witch-King.

The Witch-King had, right before the arrival of the Rohirrim, broken down the gates of Minas Tirith and was preparing to bring his army through. It was Gandalf, and only him, who stood before him and his entry into the city.

‘You cannot enter here,’ said Gandalf, and the huge shadow halted. ‘Go back to the abyss prepared for you! Go back! Fall into the nothingness that awaits you and your Master. Go!’

[...]

‘Old fool!’ he said. ‘Old fool! This is my hour. Do you not know Death when you see it? Die now and curse in vain!’ And with that he lifted high his sword and flames ran down the blade.

Gandalf did not move. And in that very moment, away behind in some courtyard of the City, a cock crowed. Shrill and clear he crowed, recking nothing of wizardry or war, welcoming only the morning that in the sky far above the shadows of death was coming with the dawn.

And as if in answer there came from far away another note. Horns, horns, horns. In dark Mindolluin’s sides they dimly echoed. Great horns of the North wildly blowing. Rohan had come at last.

_{The Lord of the Rings, The Return of the King, Chapter 4, The Siege of Gondor}

I would assume, had the Rohirrim's horns had not sounded after the cock's crow, Gandalf would have been prepared to defend himself and press an attack.

Pippin's arrival led to Gandalf's change of mind

Gandalf at least might have believed he would have to be the one to fight the Witch-King, as your first quote suggests. Denethor sums this up nicely:

‘Then, Mithrandir, you have a foe to match you,’ said Denethor.

The final telling evidence is at the start of the chapter The Pyre of Denethor, where Pippin is about to meet Gandalf. Since Gandalf didn't get to fight the Witch-King at the gates, he resolved to meeting the Witch-King in the field.

When the dark shadow at the Gate withdrew Gandalf still sat motionless. But Pippin rose to his feet, as if a great weight had been lifted from him; and he stood listening to the horns, and it seemed to him that they would break his heart with joy. And never in after years could he hear a horn blown in the distance without tears starting in his eyes. But now suddenly his errand returned to his memory, and he ran forward. At that moment Gandalf stirred and spoke to Shadowfax, and was about to ride through the Gate.

[...]

Gandalf looked through the gaping Gate, and already on the fields he heard the gathering sound of battle. He clenched his hand. ‘I must go,’ he said. ‘The Black Rider is abroad, and he will yet bring ruin on us. I have no time.’

_{The Lord of the Rings, The Return of the King, Chapter 7, The Pyre of Denethor}

Beyond hope the Captain of our foes has been destroyed, and you have heard the echo of his last despair. But he has not gone without woe and bitter loss. And that I might have averted but for the madness of Denethor.

_{The Lord of the Rings, The Return of the King, Chapter 7, The Pyre of Denethor}

The emphasised parts clearly show that Gandalf had the intention of riding out into the battle to confront the Witch-King, and possibly defeat him, thus simultaneously stopping him from doing more damage to the defenders. Only the arrival of Pippin and his news stopped him from doing so.