I will begin with this confession: whatever I have done in the course of my life, whether it be good or evil, has been done freely; I am a free agent.
The doctrine of the Stoics or of any other sect as to the force of Destiny is a bubble engendered by the imagination of man, and is near akin to Atheism. I not only believe in one God, but my faith as a Christian is also grafted upon that tree of philosophy which has never spoiled anything.
As he notes, he writes his memoirs as a confession, but not in the style of a repentant sinner; he regards his indulgences as the follies of youth, and portrays them in that light. The Preface has what looks like a sharp rejection of Cartesianism:
A certain philosophy, full of consolation, and in perfect accord with religion, pretends that the state of dependence in which the soul stands in relation to the senses and to the organs, is only incidental and transient, and that it will reach a condition of freedom and happiness when the death of the body shall have delivered it from that state of tyrannic subjection. This is very fine, but, apart from religion, where is the proof of it all? Therefore, as I cannot, from my own information, have a perfect certainty of my being immortal until the dissolution of my body has actually taken place, people must kindly bear with me, if I am in no hurry to obtain that certain knowledge, for, in my estimation, a knowledge to be gained at the cost of life is a rather expensive piece of information.
He also laments over Spinoza:
Oh, blissful ignorance! Spinosa, the virtuous Spinosa, died before he could possess it. He would have died a learned man and with a right to the reward his virtue deserved, if he had only supposed his soul to be immortal!
It is not true that a wish for reward is unworthy of real virtue, and throws a blemish upon its purity. Such a pretension, on the contrary, helps to sustain virtue, man being himself too weak to consent to be virtuous only for his own 'gratification. I hold as a myth that Amphiaraus who preferred to be good than to seem good. In fact, I do not believe there is an honest man alive without some pretension, and here is mine.
Was Casanova a freethinker? Claire is right that he certainly has views that would ordinarily be considered freethinking rather than orthodox. On the other hand, he seems to distance himself from freethinkers:
A so-called free-thinker told me at one time that I could not consider myself a philosopher if I placed any faith in revelation. But when we accept it readily in physics, why should we reject it in religious matters? The form alone is the point in question. The spirit speaks to the spirit, and not to the ears. The principles of everything we are acquainted with must necessarily have been revealed o those from whom we have received them by the great, supreme principle, which contains them all. The bee erecting its hive, the swallow building its nest, the ant constructing its cave, and the spider warping its web, would never have done anything but for a previous and everlasting revelation. We must either believe that it is so, or admit that matter is endowed with thought. But as we dare not pay such a compliment to matter, let us stand by revelation.
I think that what he chiefly has in mind here is the immortality of the soul: Casanova is a skeptical fideist about the immortality of the soul. Reason cannot prove it (pace Descartes), it must be accepted on faith alone (pace an unnamed freethinker), and it is rational to hope for it (pace Spinoza).
He has an interesting discussion with a Muslim in chapter XIV, where he implicitly accepts the Incarnation, and gives a fideist response to the Muslim:
"My religion tells me to believe without arguing, and I shudder, my dear Yusuf, when I think that, through some specious reasoning, I might be led to renounce the creed of my fathers. I first must be convinced that they lived in error. Tell me whether, respecting my father's memory, I ought to have such a good opinion of myself as to sit in judgement over him, with the intention of giving my sentence against him?"
However, after the conversation, he allows that Yusuf might be right; but still says that it would be absurd to leave the creed of his fathers for Islam because even if Muhammed were right on this point, he was still an 'arrant imposter'. (The discussions with Yusuf are all fascinating, and well worth reading.) He is certainly a Catholic (goes to confession regularly, is certain that he is absolved of his sins in it, etc.) He is severely critical of the adoration of the Sacred Heart (which he thinks a bit disgusting), and is suspicious of Jesuits, but neither of those would have been uncommon for Catholics at the time. He reminds me in many ways of Montaigne: philosophically skeptical, fideistically Catholic. His memoirs are a great resource for those interested in early modern fideism.