Israeli Prime Minister Binyamin Netanyahu accused Iran "and its protege Hezbollah" of being behind both attacks. Israel is braced for more attacks.
Iran is suspected of seeking payback for the assassination of at least four nuclear scientists in Tehran over the last couple of years.
Those killings were also blamed on Israel's vaunted foreign intelligence service, the Mossad, which the French newspaper Le Figaro reported in January was training Iranian exiles in Iraqi Kurdistan to carry out bombings in Iran to destabilize the country.
Monday's attacks brought the deadly intelligence war between Israel and Iran into sharp relief, raising tensions in the Middle East amid a simmering confrontation between Iran and the West in the Persian Gulf that could trigger a region-wide conflict.
The bombing in New Delhi blasted an Israeli diplomat's car near the Israeli embassy, wounding his wife and the driver.
In Georgia, security officials found a grenade in a black plastic bag attached to the underside a car owned by a driver at the Israeli Embassy in Tbilisi.
Monday's near-simultaneous attacks followed months of terror alerts as Israel's security authorities warned that Hezbollah was expected to mount operations aimed at avenging Mughniyeh's assassination in Damascus.
In January, there were reports in Baku, capital of the former Soviet republic of Azerbaijan, that security services there had thwarted Iranian-backed plots that were seen as retaliation for the killing of the Iranian scientists.
One reportedly involved killing teachers at a Jewish school attended by some 400 pupils near Baku. Another involved a plot to assassinate Israeli Ambassador Michael Lotem. Two of three suspects were arrested by Azeri authorities. The plotters had allegedly received $150,000 from Tehran.
The Sunday Times of London reported last weekend that the Mossad used oil-rich Azerbaijan as a base for espionage operations to spy on Iran, Azerbaijan's southern neighbor. Relations between the two countries have been strained since the collapse of the Soviet Union in 1989.
Muslim-majority Azerbaijan has moved closer to the West and Israel. Iran, which has a large Azeri population, accuses Baku of seeking to destabilize the country and take over the northern, Azeri-dominated region.
Iran's official news agency reported that Tehran responded angrily to The Sunday Times report.
"Following the movements in the Azerbaijan republic of the terrorists involved in the assassination of Iranian scientists and the facilities provided to them to go to Tel Aviv in collaboration with Mossad spy networks, Azeri Ambassador to Iran Javanshir Akhundov was summoned to the Foreign Ministry," the agency said.
In 2008, Azeri authorities foiled a plot, allegedly involving Hezbollah and Iran's intelligence service, to blow up the Israeli Embassy in a high-rise Baku building.
On Jan, 12, security authorities in Thailand arrested a suspected Hezbollah operative in Bangkok for plotting an attack against Jewish targets.
The Lebanese suspect, who also has Swedish citizenship, was indicted for possessing some 10,000 pounds of urea-based fertilizer and 10 gallons of liquid ammonium nitrate. That's enough to make several truck bombs.
A second suspect remains at large.
Also in January, Israel's counter-terrorism bureau announced it had asked authorities in several European countries to tighten security around groups of Israelis at ski resorts and hotels amid fears of terrorist attacks.
Hezbollah leaders have repeatedly warned that the Iranian-backed Lebanese movement will avenge Mughniyeh.
Israeli sources said in 2011 that as many as 10 purported Hezbollah plots had been thwarted in Europe and Asia.
In January, the Israeli military tightened security around former chief of staff Lt. Gen. Gabi Ashkenazi amid fears Hezbollah planned to target him as the fourth anniversary of the Mughniyeh assassination approached.
Mughniyeh, who until Osama bin Laden came along was the most wanted terrorist fugitive in the world, was killed by a bomb hidden in his sports utility vehicle in a highly secure sector of Damascus.
Ashkenazi, who was the chief of Israel's general staff at the time of the Mughniyeh slaying, retired from the military in February 2011. At that time he was under round-the-clock protection because he had been targeted in a Hezbollah plot uncovered in August 2009.