This looks like "The Street That Wasn't There" by Clifford Simak and Carl Jacobi, currently available as an ebook at Project Gutenberg.
It starts how you remember:
Mr. Jonathon Chambers left his house on Maple Street at exactly seven o'clock in the evening and set out on the daily walk he had taken, at the same time, come rain or snow, for twenty solid years.
The walk never varied. He paced two blocks down Maple Street, stopped at the Red Star confectionery to buy a Rose Trofero perfecto, then walked to the end of the fourth block on Maple. There he turned right on Lexington, followed Lexington to Oak, down Oak and so by way of Lincoln back to Maple again and to his home.
He didn't walk fast. He took his time. He always returned to his front door at exactly 7:45. No one ever stopped to talk with him. Even the man at the Red Star confectionery, where he bought his cigar, remained silent while the purchase was being made. Mr. Chambers merely tapped on the glass top of the counter with a coin, the man reached in and brought forth the box, and Mr. Chambers took his cigar. That was all.
And later finds the confectionary store gone:
All next day he was uneasy and as he left the house that evening, at precisely seven o'clock, he kept saying to himself: "You must not forget tonight! You must remember to stop and get your cigar!"
The street light at the corner of Jefferson was still out and in front of 816 the cemented driveway was still boarded off. Everything was the same as the night before.
And now, he told himself, the Red Star confectionery is in the next block. I must not forget tonight. To forget twice in a row would be just too much.
He grasped that thought firmly in his mind, strode just a bit more rapidly down the street.
But at the corner he stopped in consternation. Bewildered, he stared down the next block. There was no neon sign, no splash of friendly light upon the sidewalk to mark the little store tucked away in this residential section.
And ends with him alone in a room contemplating whether, when he goes as everything else must, if he'll merely be a 'thing' (not quite a chair).
But, after all, what was the use? One man, alone, could not stand off the irresistible march of nothingness. One man, all alone, simply couldn't do it.
He wondered what the elephant ash tray looked like in that other dimension. It certainly wouldn't be an elephant ash tray nor would the radio be a radio, for perhaps they didn't have ash trays or radios or elephants in the invading dimension.
He wondered, as a matter of fact, what he himself would look like when he finally slipped into the unknown. For he was matter, too, just as the ash tray and radio were matter.
He wondered if he would retain his individuality ... if he still would be a person. Or would he merely be a thing?
There was one answer to all of that. He simply didn't know.
It was collected with "The Worm" in the 1948 anthology "Strange Ports of Call" and later again with it in 1975's "Creatures from Beyond". Sometimes it is titled "The Lost Street" instead, and it has been apparently revised a couple times and so specific details might be different between the two stories and the Project Gutenberg version, but maybe you can track down the specific anthology and find the exact story you remember.
