No. In the common law world, conviction of a criminal offence can only occur after a judicial process which starts with a charge laid by the prosecution. The decision to lay a charge is never automatic and involves the exercise of prosecutorial discretion.
The other answers have explained the privilege against self-incrimination. While this is an important privilege, it is not uncommon for witnesses to waive it and give incriminating evidence. For example, an accomplice may give evidence against a co-accused in the hope of receiving a lighter sentence, even if that evidence is not important enough for the authorities to induce it with an immunity from prosecution. Or a defendant facing serious charges might explain apparently inculpatory evidence by admitting responsibility for a less serious offence which has not been charged and is not available as an alternative verdict. As suggested in the question, the admission might also be incidental and not understood by the witness to be incriminating.
If the authorities choose to prosecute the witness, what the witness said in the previous court proceedings is an admission against interest. It is admissible in the same way as a voluntary confession outside the courtroom. Although the fact that it was made on oath may mean that it carries more weight, it is not conclusive. The witness might choose to give different evidence in their defence – “I was lying then, but I’m telling the truth now.”
The exercise of prosecutorial discretion depends on the overall strength of the case against the witness, including the evidence of their previous admission. It would also depend on the seriousness of the admitted offence and any applicable statute of limitations. A less serious or historical offence is unlikely to be of interest to the parties in court and may never come to the attention of the prosecuting authorities. More serious offences are likely to be referred to the police for further investigation and prosecution.