Iranian state television showed footage of two Qadr missiles being launched from northern Iran, which the Islamic Revolutionary Guards Corps (IRGC) said hit targets 1400km away.
Iranian agencies said the missiles were stamped with the Hebrew words, "Israel should be wiped from the pages of history", though the inscription could not be seen on any photographs.
Clinton, a former secretary of state under President Barack Obama, said she was "deeply concerned" by the tests, the second round of Iranian missile launches in two days.
"Iran should face sanctions for these activities and the international community must demonstrate that Iran's threats toward Israel will not be tolerated," said Clinton, who is ahead in the race to be Democratic nominee at the November 8 presidential elections.
Her call for sanctions reflected a tougher line against Iran's recent missile activity than that taken so far by the White House, which said it is aware of and reviewing reports of the Iranian tests, and would determine an appropriate response.
The Iranian move on Wednesday came despite warning from the US State Department after Tuesday's missile tests that Washington continues to "aggressively apply our unilateral tools to counter threats from Iran's missile program", a possible reference to additional American sanctions.
Secretary of State John Kerry spoke on Wednesday with Iran's foreign minister about the test-firing of two ballistic missiles, a State Department spokesman said.
The missile tests underline a rift in Iran between hardline factions opposed to normalising relations with the West, and President Hassan Rouhani's relatively moderate government, which is trying to attract foreign investors to Iran.
Iran's IRGC said the missiles tested on Wednesday were designed with Israel in mind.
"The reason we designed our missiles with a range of 2000km is to be able to hit our enemy the Zionist regime from a safe distance," Brigadier General Amir Ali Hajizadeh was quoted as saying by the ISNA agency. The nearest point in Iran is around1000km from Tel Aviv and Jerusalem.
Israeli Defence Minister Moshe Yaalon told Israel Radio the tests showed Iran's hostility had not changed since implementing a nuclear deal with world powers in January, despite Rouhani's overtures to the West.
"To my regret there are some in the West who are misled by the honeyed words of part of the Iranian leadership while the other part continues to procure equipment and weaponry, to arm terrorist groups," Yaalon said.
