Why was neither David nor Solomon stoned to death?

There needed to be two or three witnesses for such a one to be pronounced guilty and sentenced to death; one witness was not enough for the death penalty

Numbers 35:30

Whoever kills a person, the murderer shall be put to death on the testimony of witnesses; but one witness is not sufficient testimony against a person for the death penalty (nkjv).

Deuteronomy 17:6

Whoever is deserving of death shall be put to death on the testimony of two or three witnesses; he shall not be put to death on the testimony of one witness (nkjv).

Deuteronomy 19:15

"One witness shall not rise against a man concerning any iniquity or any sin that he commits; by the mouth of two or three witnesses the matter shall be established (nkjv).

Who would be the witnesses in the case of David?

None of these answers address the apparent conflict of a just king who is above the law and not accountable to its penalties- even to God. They also seem to underestimate the commitment of the Hebrew people to justice and impartiality of the law (we can't conclude that the courts contemporary to David's rule were corrupt), and disregard that the law was both a spiritual agreement with God and a legally binding criminal code within their society.

The answer in David's case is that the law condemned his actions as worthy of death, but did not allow for his execution. We can't just go around stoning people and claim that it's justified by the law. A legal system is unjust that does not permit due process and allows judgement and sentencing of death to be carried out without a trial. Thus, God gave specific instructions concerning a sentence of death in the Torah. The traditional interpretation (Babylonian Talmud, Sanhedrin 2a-b) of those instructions maintain that even an ox that is worthy of the death penalty (eg. for killing a person) should be tried by a court of 23 judges.

According to this Judaism.SE question, in order to be eligible under the law for the death penalty on either charge, David would have had to continue in his sin after being confronted by more than one witness that could testify in a court of 23 judges that he saw David persist in his sin after being confronted. This is the very reason for the observation in a previous answer that "not many people were put to death." According to the linked question, a court was considered murderous if it executed a person more frequently than 7 years (some accounts say 70 years).

Nathan, filled with the word of God, convinced David with a parable that he was worthy of death. David does not continue in his sin. He repents Uriah's murder and legitimizes his relationship with Bathsheba, and so he is not eligible to be sentenced to death for his sins (crimes).

It's interesting that in the case of David, he was actually stoned for being a murderer - it just happened a fair bit after the event and wasn't a successful execution because only one guy got in on the action:

⁵ As King David approached Bahurim, a man from the same clan as Saul’s family came out from there. His name was Shimei son of Gera, and he cursed as he came out. ⁶ He pelted David and all the king’s officials with stones, though all the troops and the special guard were on David’s right and left. ⁷ As he cursed, Shimei said, “Get out, get out, you murderer, you scoundrel! ⁸ The Lord has repaid you for all the blood you shed in the household of Saul, in whose place you have reigned. The Lord has given the kingdom into the hands of your son Absalom. You have come to ruin because you are a murderer!”

⁹ Then Abishai son of Zeruiah said to the king, “Why should this dead dog curse my lord the king? Let me go over and cut off his head.”

¹⁰ But the king said, “What does this have to do with you, you sons of Zeruiah? If he is cursing because the Lord said to him, ‘Curse David,’ who can ask, ‘Why do you do this?’”

¹¹ David then said to Abishai and all his officials, “My son, my own flesh and blood, is trying to kill me. How much more, then, this Benjamite! Leave him alone; let him curse, for the Lord has told him to. ¹² It may be that the Lord will look upon my misery and restore to me his covenant blessing instead of his curse today.”

¹³ So David and his men continued along the road while Shimei was going along the hillside opposite him, cursing as he went and throwing stones at him and showering him with dirt. ¹⁴ The king and all the people with him arrived at their destination exhausted. And there he refreshed himself. - 2 Samuel 16:5-13 NIV

It seems that David didn't challenge in any way the legitimacy of this treatment, but instead asserted that Shimei was following the Lord's will. Rather than excusing himself from such treatment, David trusted in God's mercy to rescue him from what was a righteous judgment.

This question isn't really a Christian doctrinal question, I think. But, the most obvious answer here is that they were both kings. And the literal letter of God's law is always enacted by people (like the King's army or guards), who are generally under the rule of the king -- notably as a sort of proxy for God in the case of the Jews.

So, the king probably has guards, for one. And these guards probably don't question the king's actions or authority. The people, therefore, are not likely going to form a clan and go up against the king's armed guards because he slept around. They'd just be slaughtered.

I.e., no one's going to stone the king unless God explicitly tells the people to do so via a prophet, even if the king should rightfully be stoned according to the letter of the law! And even then, the king's guards probably need to be in cahoots with the rebels, so to speak, before the people will actually take action -- or at least any successful action.

So, regardless of whether there was a good or strictly legal reason that David and Solomon shouldn't have been stoned, there's no practical reason they would have been. And there are plenty of practical de-motivators at work.

Yes, there was a special provision. God personally enacted a punishment.

In the case of David, God caused his son to die and did not permit him to be the one to build the temple.

2 Samuel 12:14 (ESV)
Nevertheless, because by this deed you have utterly scorned the Lord, the child who is born to you shall die.

1 Chronicles 28:3 (ESV)
But God said to me, ‘You may not build a house for my name, for you are a man of war and have shed blood.’