It is narrated from Sayyiduna Ibn Abbas (may Allah be pleased with them) that he said: I consider replying to a letter as necessary as replying to a greeting of peace (salam).
Hadith Referenceالادب المفرد / كتاب الرسائل / 1117
Hadith Gradingالألبانی:حسن
Hadith Takhrij«حسن : أخرجه ابن أبى شيبة : 26369 و ابن الجعد : 2399 و البيهقي فى شعب الإيمان : 9097»
Explanation & Benefits
Shaykh Maulana Usman Muneeb
Benefits and Issues: The intent of Sayyiduna Ibn Abbas (radi Allahu anhuma) is that just as replying to the greeting of salam is obligatory, in the same way, replying to a letter is also obligatory.
Source: Fadlullah al-Ahad: Urdu Commentary on al-Adab al-Mufrad, Page: 1117