In a speech broadcast by RTE, Kenny said he had "no intention of being diverted from... that responsibility that I've undertaken and which I have received a mandate to fulfil".
At an event in Castlebar, Co Mayo, Mr Kenny ruled out an early departure and said his focus had always been on securing Ireland's future.
Merkel and Kenny said they expected the new United Kingdom government under Theresa May, which takes office on Wednesday, to quickly define its relationship with the EU.
Merkel reiterated that it was now up to London to formally trigger Article 50 to leave the EU following last month's shock referendum backing a "Brexit" or British exit from the Union. The 27 member states will bring their influence to bear. It is important that Ireland plays a part here and we will do so in a special friendship.
"The EU could not put forward any proposals on the two sides' future relationship until it's clear what the strategy from Britain would be", said Kenny following talks in Berlin with Merkel.
He said: "We have an agreement with Fianna Fail in terms of confidence and supply, we have a programme for government with the independent alliance and the independents with over 600 commitment and my focus is entirely on that future in making this happen".
"Yet I am firmly convinced that the European Union is strong enough to survive this turning point", she added.
His comments come after a backbencher in his party called for a new leader to be in place by September.