After rejecting the offer of 215p a share as undervaluing McAlpine, Whitehead said he would not "throw the baby out with the bathwater" by changing his business strategy.
Fund manager Phillips & Drew, which owns 23% of McAlpine, backed the offer that was made by an unnamed individual. Almost all his funding for the £240m offer was provided by Royal Bank of Scotland.
Analysts said that if the bid had been accepted, the buyer would have split and sold McAlpine's contracting and housebuilding arms.
But Whitehead argued that to change strategy by demerging McAlpine's businesses would destroy the current management's credibility. It promised shareholders as recently as March that earnings per share would grow 15% a year for the next three years .
Besides, Whitehead said, he is not sure that the separated businesses would have "sufficient clout" to operate individually.
Analysts said they believe the offer for McAlpine could tempt other housebuilders to launch a bid for the company with a view to selling off or closing the contracting arm. Cash-rich Beazer and Persimmon were touted as possible bidders.
Analysts said the least likely scenario was that a bidder would buy McAlpine for its contracting arm alone – there are now plenty of contractors to buy without the additional complication of a housebuilding interest.
They also said relationships between the McAlpine management and Phillips & Drew would be strained by the investor's resounding vote of no confidence in it.
Whitehead said: "The fact that one major shareholder accepted this offer cannot be ignored, but there are still another 77% of the shares they do not speak for. We will talk to our other shareholders to explain our plans and see what they say." Phillips & Drew has in recent months been attempting to cajole the management at those construction companies in which it has a major stake to take radical action to increase shareholder value.
But one analyst said that backing the mystery individual seemed a "strange" move for Phillips & Drew, as the McAlpine management has been relatively successful and was halfway through a three-year recovery plan. This could eventually see strong dividend growth, he said.
It is now possible that Phillips & Drew, having forced a 37.5p rise to 230p in McAlpine's share price, might sell its stake in dribs and drabs.
Eliminating such a large holding in the company would take pressure off Whitehead, said the analysts.