Manor Care Inc. has promoted C. Arnold Renschler, president of the Silver Spring firm's leading health-care subsidiary, to president and chief operating officer of this diversified owner and operator of nursing homes and hotels.

Renschler, 43, replaces Stewart Bainum, the 66-year-old founder of Manor Care. Bainum, who with his family owns about 37 percent of Manor Care's stock, will remain the firm's chairman and chief executive officer with responsibility for corporate development.

The appointment is a "vote of confidence" for Renschler, who has supervised a three-year period of unprecedented growth by the firm's largest subsidiary, Manor Healthcare Corp., said company spokesman Ray Murphy. Manor Healthcare, which is responsible for about 80 percent of the parent company's assets and 90 percent of its earnings, has grown through acquisitions and innovative marketing from a $165 million operation in 1982 to $357 million this year. Renschler will continue as president of this division.

Earlier this month, the parent company reported record earnings of $30.2 million on revenue of $454.4 million in fiscal 1985, up 32 percent from the previous year. Currently, Manor Care is the second-largest publicly owned nursing home company in the country.

Renschler, a doctor, served on Manor's board of directors and as a vice president for corporate development for seven years. Earlier, he operated a private medical practice and was on the faculty of John Hopkins University School of Medicine and Stanford University Hospital.