Sheikh `Abd al-Wahhâb al-Turayrî, former professor at al-Imâm University in Riyadh
Not exactly. "Capital punishment" would mean that the state is obliged to carry out the death penalty against the murderer if found guilty of the crime, regardless of the wishes of the victim's family. What Islam has prescribed for murder is that it has given the relatives of the victim the option of seeking retribution.
Allah says: “O you who believe! The law of retribution is prescribed for you in cases of murder.” [Surah al-Baqarah: 178]
The relatives of the victim have three options open to them. They may pardon the killer. They may opt to receive monetary compensation in the form of blood money(diyah). They may request retribution (qisas).
The Prophet (peace be upon him) said: “Whoever has one of his relations murdered has the choice either to receive compensation thereto or have the killer killed.” [Sahîh al-Bukhârî (2434) and Sahîh Muslim (1355)]
This is done before a proper court of law. If the relatives of the deceased opt for retribution, then the court is obliged to carry out the death pealty on their behalf. Otherwise, they have no jurisdiction to execute the murderer.
Source: Islam Today