Mr Jammeh told state TV the proclamation was in line with Gambia's "religious identity and values".
He added that no dress code would be imposed and citizens of other faiths would be allowed to practise freely.
Some 90% of Gambians are Muslim. The former British colony's economy relies heavily on tourism.
However, relations with the West have soured recently.
The European Union temporarily withheld aid money to Gambia last year over its poor human rights record.
Mr Jammeh has been president of the tiny West African country for 21 years.
"As Muslims are the majority in the country, the Gambia cannot afford to continue the colonial legacy," Mr Jammeh told state TV, explaining his decision to proclaim an Islamic republic.
Other Islamic republics include Iran, Pakistan and - in Africa, Mauritania.
Mr Jammeh withdrew Gambia from the Commonwealth in 2013, describing the organisation as neo-colonial.
In 2007, he claimed to have found a herbal cure for Aids.

The new measure comes into force on Friday as police launch their annual winter drink-drive campaign.
Previously, police needed reasonable suspicion before asking a driver to take a breath test.
The new legislation was passed through the Assembly in April, but this is the first time it has been used by police.
The PSNI said there was no safe limit and that motorists should never drink and drive.
Assistant Chief Constable Alan Todd said the new legislation gave police "another tool that will hopefully help us to prevent people taking life-threatening, unacceptable, simply stupid risks".
"Previously, police officers needed a reasonable suspicion about the manner of someone's driving, have seen a moving traffic offence, or been called to a collision before requiring a preliminary breath test from a driver," he said.
"The new legislation means we can now establish vehicle checkpoints solely for the purpose of carrying out random breath tests, something which we hope will act as an even more visible, physical deterrent."
During last year's operation, almost 400 people were caught drink-driving by police.
"Just one drink can impair ability to drive. Considering that in some instances, we have stopped drivers who were so drunk, they could barely stand when they got out of their vehicle, just beggars belief," said Ass Chief Const Todd.
"At the other end of the spectrum, we detected some drivers who had gone out socialising and not intended to drive, but their circumstances changed and they decided to take a risk. A risk which inevitably results in a driving ban."
Police said they would also be working with officers from the Irish police traffic department in border counties.
"In addition to the checkpoints, any driver or motorcyclist we stop, whether for speeding, using a mobile phone, or committing any moving traffic offence can expect to be breathalysed," he added.
"So too can anyone involved in a collision or who we suspect may have consumed alcohol or taken drugs."
Gambian President Yahya Jammeh has declared his Muslim-majority country an Islamic republic, saying the move marks a break with the colonial past.
[NEXT_CONCEPT]
Police are to conduct random breath tests at vehicle checkpoints as a result of new legislation in a bid to crack down on drink-driving.