Yes, Rama did exile Sita. This story is narrated in the Uttara Kanda of Valmiki Ramayana.
The reason for the exile can be traced to this particular conversation Rama has with one of his counselors, Bhadra, who upon Rama's request gives an account of both good and bad things the people of Ayodhya generally speak of his actions:
Hearing the words of Bhadra, Rāma said :– "Without hiding anything, do
you relate everything from the beginning vent to by the citizens.
Hearing the good and bad opinions of the citizens I shall desist from
bad actions and engage in good ones."...
Hear, O king, I shall relate to you all those unpleasant things frequently dwelt upon by people in court-yards, markets, public roads, forests and gardens...
Rāvana did forcibly place Sitā on her lap; how can then Rāma enjoy
delight in her company? Having taken her to the city of Lankä, Rāvana
did keep her in the Aśoka forest and Sitā was brought under the
control of Ráksasas. Still Rāma has not been worked up with hatred by
Sitā. From now we shall also brook the bad conduct of our wives — for
the subjects always tread the footsteps of their King.
[Uttara Kanda, Chapter 43]
Hearing those words, Rama contemplates and then later explaining his reasons instructs Lakshmana to leave Sita at Valmiki's ashram:
Hear, what the citizens have been talking about me and Sītā. The citizens as well as the inhabitants of provinces have been showering censures upon me...you know how in the solitary forest of Dandaka, Sitā was stolen away by Rāvana and how have I slain him. At that time even I was stricken with anxiety regarding Sītā that how I could take her home since she had resided in the house of the Rākşasas.
To secure my confidence, Sitā, in your very presence, entered fire. At that time, O Saumitri, fire, carrying sacrificial oblations and the wind of the sky declared Sītā’s innocence before the celestials. in the presence of all the Rsis and Do you, O Laksmana, next morning Indra, the king of the celestials, himself handed over the chaste Sitä to me in the island of Lańkā. My mind know Sītā as chaste for ever. So, at that time, I came back to Ayodhyá with Sītā.
But now a great sorrow consequent upon the censure of the citizens and villagers has pierced my heart. He, who is notorious on this earth and as long as that notoriety remains current, is classed among the vile. Even the celestials speak ill of bad name — whereas fame is adored in all the regions. Therefore the high-souled exert their best to acquire reputations. O foremost of men, what to speak of the daughter of Janaka — I can even renounce my life and yourselves in fear of a bad name.
Do you therefore perceive into what great abyss of sorrow and ill-fame I have fallen. Up to this time I have never experienced such a mighty grief.
Ascending the car driven by Sumantra, take away SItă to another country.
There is a picturesque hermitage of the high-souled Vălmîki situated on the Tamasă on the other side of the river Gangå. Do you, O delight of Raghus, soon come back, leaving behind Sītā in that lonely place. Do you carry out my words. Do you not speak anything regarding Sītā's banishment...
Ere this Sītā had communicated her intention of beholding the hermitages of ascetics on the banks of the Gañgå. Let that desire of hers be now satisfied.
While saying this, the eyes of the virtuous-souled Rāma were covered with tears. Sighing hard like to an elephant, he, with a heart stricken with grief, departed to his own quarter in the company of his brothers.
[Uttara Kanda, Chapter 45]
Towards the end of Ramayana, Rama does make one last attempt to prove Sita's chastity in front of everyone with the help of Valmiki's attestation:
And understanding from the story that Kuša and Lava were Jānaki's sons, Rāma mentioning her name said before the assembly:- Send a good emissary to the illustrious Välmiki and let him communicate to the ascetics that if Jānaki is sinless and has lead a pure life in the forest; let her give proof of purity by the great ascetic’s permission.
Let the emissaries learn well the intention of the ascetic in this and if SItā is at heart willing to bring in proofs. To uphold her as well as mine purity, let Maithili, the daughter of Janaka, swear before the assembly in the morning of the next day.
Hearing the words of Rāma, messengers went to Vālmīki and saluting the high-souled (ascetic) burning in his effulgence and of incomparable lustre, communicated to him, in sweet words all what Rāma had said. Hearing the words of the messengers and understanding Rāma’s intention the ascetic said. What Rāma has said shall be satisfied. May good betide you. Husband is the greatest god for women. So Sita shall carry out his behests. The great ascetic having said this, the highly powerful emissaries, approaching Räma, communicated to him what the Muni had said. Hearing the words of the high-souled Vālmīki, Rāma was greatly delighted.
[Uttara Kanda, Chapter 95]
The next morning, Valmiki testifies before Rama's assembly:
There upon having entered that huge assemblage in company of Sitā, the
great ascetic Vālmīki addressed Rāma, saying: O son of Dašaratha, in
fear of calumny, you did renounce her near my hermitage, albeit Sitā
is pure and follows the ways of morality.
O Rāma of firm vows, you are afraid of the censure of the people. SItā
has becoming willing to give testimony (of her purity) for removing
the calumny of the public; do you mercifully give her permission in
this.
I tell you the truth, O Rāma, that these irrepressible twin brothers
are your sons, O descendant of Raghu. I am the tenth son of Pracetă,
so, far from speaking untruth it does not even spring up in my mind. I
therefore know the truth that these twins are your sons. I have
performed austere penances for many thousand years; I now swear before
you, that if this Maithili is found touched by any sin I shall not
reap the fruit of my ascetic observances extending over many thousand
years.
I have never perpetrated before a crime either, in my mind, body, or
speech. If Maithili is found divested of sin I may then partake of the
fruits of piety.
O descendant of Raghu, finding Sītā pure in mind and five elements of
body I took her near the spring in the forest.
Besides this Sītā, of pure ways, divested of sins and ever considering
her husband as deity, shall give testimony this day.
You have been afraid of the public calumny. O Son of a king, your mind
being possessed by the fear of public scandal regarding her husband as
a god; I came to know this by virtue of my discriminative knowledge.
[Uttara Kanda, Chapter 96]
Rama then accepting Kusa and Lava as his sons, apologizes to Valmiki:
O Brähmana, vilification of the people is very powerful. Although I know that Jānaki has not been touched by any sin still I have renounced her. Do you therefore forgive me.
I know that these twin brothers Kuša and Lava are my sons. Yet if Jānaki gives evidence of her own purity before the assembly I shall be greatly satisfied.
[Uttara Kanda, Chapter 97]
Now although Sita testifies as Rama wanted, the result is not what he had expected because as part of her testimony, Sita is also praying for her mother Vasundharā ([Bhū-Devī][6]) to take her back.
Thereupon beholding all people assembled there, Janaka's daughter,
wearing a red cloth, with her face and looks downwards and folded
palms said.
I have never thought of any other person in my mind but
Rāma, by the strength of this virtue let the goddess Vasundharā give
me room.
I have always with my mind, body and words prayed for Rāma's
well-being and by virtue hereof may the goddess Vasundharā give me
room in her womb.
I have never thought for other than Rāma. If it is true, may the
goddess Vasundharā give me room in her womb.
As soon as Sītā swore in this wise a wonderful incident took place.
From inside the earth a celestial and excellent throne rose up. It was
carried on head by the Nāgas of unmitigated prowess, having celestial
persons and adorned with celestial jewels and ornaments.
Having stretched out her arms and taken Maithili, the goddess earth
welcomed her and placed her on the throne. And while seated on the
throne she was entering the earth, she was covered with the continued
showers of celestial flowers...Beholding Sitā's censure thus removed all
animals, animate or inanimate, either on earth or in the sky, were
greatly delighted.
[Uttara Kanda, Chapter 97]
