Mohammed Abdelkarim, whose mother and sisters were inside the house when it was set alight, says Hebrew slogans were scrawled on the house.
Four rockets were fired in the past 24 hours, the military said, without elaborating on the test or type of rockets fired.
Streets are subdued, marketplaces are quiet and people are on edge in Jewish areas of Jerusalem.
The house demolished in the Silwan neighborhood near the Old City belonged to Abdel Rahman al-Shaludi.
UN Secretary General Ban Ki-moon strongly condemned an attack on a synagogue in West Jerusalem.
Israeli authorities gave preliminary approval for construction of 200 new homes in a Jewish area of east Jerusalem.
Palestinian President Mahmoud Abbas accused Israel on Tuesday of leading the region toward a “religious war”.
Palestinians have repeatedly clashed with police at the site.
Palestinians are urging UNSC to demand that Israel stop incitement against Muslim holy sites in Jerusalem.
Amnesty International says Israel displayed “callous indifference” in attacks on family homes in the densely populated coastal strip.
Israel recalled its ambassador to Sweden for “consultations”, hours after Stockholm recognised Palestine.
The UN Security Council will hold an emergency meeting on recent tensions between Israel and the Palestinians.
UN chief Ban Ki-moon urged Israel and the Palestinians on Tuesday to move away from unilateral actions.
The location is the holiest site in Judaism and the third holiest in Islam, and is a frequent flashpoint for demonstrations.
Israel foreign minister said he “regrets” that Sweden “rushed” to make its decision without properly understanding the conflict.