Answer by taking a quote from the following article:

Sayyid Ruhollah Musavi Khomeini (Persian: syd rwHllh mwswy khmyny [ru:hol'la:he khomei'ni:] ( listen); 24 September 1902 - 3 June 1989), known in the Western world as Ayatollah Khomeini, was an Iranian Shia Muslim religious leader and politician. He was the founder of Iran as an Islamic republic and the leader of its 1979 Iranian Revolution that saw the overthrow of 2500 years of Persian monarchy and Mohammad Reza Pahlavi, the last Shah of Iran.

In January 1963, the Shah announced the "White Revolution", a six-point programme of reform calling for land reform, nationalization of the forests, the sale of state-owned enterprises to private interests, electoral changes to enfranchise women and allow non-Muslims to hold office, profit-sharing in industry, and a literacy campaign in the nation's schools. Some of these initiatives were regarded as dangerous, especially by the powerful and privileged Shi'a ulama (religious scholars), and as Westernizing trends by traditionalists (Khomeini viewed them as "an attack on Islam"). Ayatollah Khomeini summoned a meeting of the other senior marjas of Qom and persuaded them to decree a boycott of the referendum on the White Revolution. On 22 January 1963 Khomeini issued a strongly worded declaration denouncing the Shah and his plans. Two days later the Shah took an armored column to Qom, and delivered a speech harshly attacking the ulama as a class.  Khomeini continued his denunciation of the Shah's programmes, issuing a manifesto that bore the signatures of eight other senior Iranian Shia religious scholars. In it he listed the various ways in which the Shah had allegedly violated the constitution, condemned the spread of moral corruption in the country, and accused the Shah of submission to the United States and Israel. He also decreed that the Nowruz celebrations for the Iranian year 1342 (which fell on 21 March 1963) be canceled as a sign of protest against government policies.  On the afternoon of 'Ashura (3 June 1963), Khomeini delivered a speech at the Feyziyeh madrasah drawing parallels between the Sunni Muslim caliph Yazid, who is perceived as a 'tyrant' by Shias, and the Shah, denouncing the Shah as a "wretched, miserable man," and warning him that if he did not change his ways the day would come when the people would offer up thanks for his departure from the country.  On 5 June 1963 (15 of Khordad) at 3:00 am, two days after this public denunciation of the Shah Mohammad Reza Pahlavi, Khomeini was detained in Qom and transferred to Tehran. This sparked three days of major riots throughout Iran and led to the deaths of some 400. That event is now referred to as the Movement of 15 Khordad. Khomeini was kept under house arrest and released in August.

When did he start Opposing the White Revolution?
On 22 January 1963 Khomeini issued a strongly worded declaration denouncing the Shah and his plans.