Dale said that it was "understandable" that some listeners were angry at the departure of Whale, but asked them to "have the good grace to give [him] a fair chance".
Tough-talking Whale, 61, yesterday confirmed his departure from the London-based radio station after five years, claiming that his contract had not been renewed.
LBC has now announced a "new look" for its weekday evening schedule, involving Dale taking over the weekday Drivetime programme (Monday to Friday, 4pm to 8pm).
Known for regular political blogging and commenting, conservative-minded Dale has also presented discussion shows on LBC 97.3 since September 2010.
He accepted that any major presenter change is "always difficult", but urged listeners to give him a chance.
"I'm thrilled to be given this opportunity and hopefully I can build on what James Whale has achieved in his four years on the show," he said in a blog post.
"I'm really sad to see James go as he has been fantastic to me and become a good friend since I joined LBC in September 2010.
"We have a very different style, but I have learned so much from him. He is one of the iconic broadcasters of the last three decades and I hope we see him back on the radio before too long."
Dale added: "Any change in presenter is always difficult and James has a huge following, so I am well aware that I have big shoes to fill.
"All I ask is that those who are understandably lamenting James's departure all over Twitter and elsewhere will have the good grace to give me a fair chance."
Also in the new look LBC evening schedule, long-standing presenter on the station Clive Bull returns to the weekday evenings slot (Monday to Thursday, 8pm to 10pm).
James Rea, LBC 97.3's managing editor, said of the changes: "LBC 97.3 has always been home to some of the country's best broadcasters and I know that Iain and Clive will help drive the station forward in this, our 40th birthday year."
LBC 97.3 broadcasts in Greater London on 97.3 FM and across the UK on digital and online.