Please see What are the Islamic rulings on Rape? for the punishment of rape under Islamic law. The punishment for Rape can be stoning to death if the culprit is 'muhsan' (is married or was married in the past).
As for stoning regardless of the marital status, then this hadith is not enough evidence for that since it is possible that the culprit was known to be married and that the specification of that is omitted by the narrator. Just like the mention of marital status is omitted in some versions of the hadith about the stoning of Ma'iz and the Ghamidiyya woman. Commentators on the hadith of the rape have noted that the culprit was married:
ارجموه - أي فرجموه لكونه محصنا
— Awn al-Mabood Sharh Sunan Abi Dawood
ارجموه - ومعناه أنه أقر بالزنا، فأمر برجمه، فرجموه لكونه محصنا
— Sharh Mishkaat al-Masaabeeh
Further, there are athar on the judgements of the caliphs regarding cases of rape, and their verdicts corroborate the idea that the punishment is the same as that of Zina, i.e. lashing and exile for the unmarried and that stoning is carried out only if the culprit is muhsan:
أن رجلا أضاف أهل بيت، فاستكره منهم امرأة، فرفع ذلك إلى أبي بكر فضربه ونفاه، ولم يضرب المرأة
A man was invited for a meal to a home. He then forced a woman of theirs to have sexual relations with him. When this case was brought to Abu Bakr he had the man flogged and exiled and did not punish the woman.
— Musannaf Ibn Abi Shaybah , Sunan al-Kubra Bayhaqi , Musannaf Abdur Razzaq
أن عبدا من رقيق الإمارة وقع على وليدة من الخمس، فاستكرهها حتى افتضها، فجلده عمر الحد ونفاه، ولم يجلد الوليدة من أجل أنه استكرهها
A governmental male-slave tried to seduce a slave-girl from the Khumus of the war booty till he deflowered her by force against her will; therefore 'Umar flogged him according to the law, and exiled him, but he did not flog the female slave because the male-slave had committed illegal sexual intercourse by force, against her will.
Additionally, the following authentic hadith forbids killing a Muslim except in three cases:
لا يحل دم امرئ مسلم يشهد أن لا إله إلا الله وأني رسول الله إلا بإحدى ثلاث النفس بالنفس والثيب الزاني، والمارق من الدين التارك الجماعة
The blood of a Muslim who confesses that none has the right to be worshipped but Allah and that I am His Apostle, cannot be shed except in three cases: In Qisas for murder, a married person who commits illegal sexual intercourse and the one who reverts from Islam (apostate) and leaves the Muslims.
And rape by an unmarried person is not among the three cases. Any exceptions that are made to this rule must be based on strong evidence, the hadith which you are relying on is not enough for that.