Yes, it is permissible.
سَالِمًا, يُحَدِّثُ عَنْ أَبِيهِ , قَالَ: قَالَ النَّبِيُّ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ: لَا يَمَسَّ الْقُرْآنَ إِلَّا طَاهِرًا
Salim narrated from his father that the Prophet ﷺ said, ‘None except the pure can touch the Quran.’ (Sunan Darul Qutni, 437).
Although it is recommended to maintain a state of Wudu at all times, and especially when engaging in Dhikr, it is not obligatory to have Wudu for Dhikr (or Darood). Therefore, it is permissible to recite Darood (Salah ala al-Nabi) without Wudu and even touch the paper or surface on which it has been written. So it is certainly permitted to touch a mobile phone screen under which the Darood text appears.
In the Hadith above, which is authentic when considering all narrations (see link below), we see that there is a requirement to have Wudu when touching the Holy Quran, however that requirement does not extend to other religious text. Therefore, it is not possible to consider Wudu obligatory for other religious text, but a ruling of recommendation is possible especially since it is generally recommended to maintain a state of Wudu.
فَالْوُضُوءُ لِمُطْلَقِ الذَّكَرِ مَنْدُوبٌ
فَقَدْ نَصَّ فِي أَذَانِ الْهِدَايَةِ عَلَى اسْتِحْبَابِ الْوُضُوءِ لِذِكْرِ اللَّهِ تَعَالَى. (رد المحتار على الدر المختار)
Imam al-Haskafi: Wudu is generally recommended for Dhikr.
Imam Ibn Abideen: It is explicitly mentioned in the Adhan section of al-Hidayah that Wudu is recommended for the Dhikr of Allah Most High. (Radd al-Muhtar).
And Allah Most High knows best.
– Answered by Shaykh Noorud-deen (24.05.2021)
Is it permitted to touch a copy of the Holy Quran without Wudu?
Hadith about specific reward for reciting Darood Sharif on Friday
See also (video):