Just to clarify, it seems the Conservative Party and DUP likely didn't form a coalition, but rather operate on a "confidence and supply" agreement, as reported in The Guardian.
There has been no decision as yet on whether there will be a formal coalition between the Conservatives and the DUP or if they will operate on a “confidence and supply” arrangement – whereby the unionists would support a minority government on vital matters in return for some of their policies being enacted.
It has been reported that the two parties do not believe it necessary to enter a formal coalition to govern.
As to whether a new general election will be called will depend on the new Prime Minister. It's also legally not required for an unelected PM to call a general election and there have been many former Prime Ministers who assumed office unelected and did not call for any election.
The most recent example would be Gordon Brown, who won the Labour leadership election unopposed and didn't call for any election, thus remaining an unelected PM.
In fact, there are 2 other Prime Ministers after WWII who were unelected, they are Lord Home and James Callaghan.