Solih took the oath office with thousands watching on at the national stadium in Malé.
More than 300 foreign dignitaries from 46 countries were expected to attend the inauguration ceremony.
Yameen said he was respecting the wishes of thousands of supporters.
The four leaders met for the first time since September's election victory.
Former speaker Shahid is expected to join the cabinet.
Transition committees have been assessing ministries and state-owned enterprises.
The draconian law was abolished with 38 votes in favour and 13 against.
A 16 percent drop in the wage bill was promised with 100 fewer political appointees.
The Dhiggaru MP was approved with 67 votes in favour and one abstention.
The resolution was passed with 57 votes in favour from the 85-member parliament.
The widely-condemned law introduced hefty fines and jail terms for journalists.
A no-confidence motion was passed with 42 votes in favour and 24 votes against.
Nazim returned as a free man after the Supreme Court suspended his 11-year jail sentence.
It would not be a bilateral visit and Modi will return to New Delhi on the same night.
Niyaz drew fire from the former ruling lawmakers in a Twitter spat last week.
Reports suggest there will be 22 ministries, up from 14 ministries in the outgoing administration.
Solih will meet with Saudi government officials during the trip.
