You made changes to your update statements to merge them into one, but got lost with some logic.

Update Asgnmnt

Set DateAcknow =@ClosedDate,

TimeAcknow=@ClosedTime,

WhoAcknow =@ClosedBy,

DateResolv = @ClosedDate,

TimeResolv = @ClosedTime,

WhoResolv = @ClosedBy

Where CallId = @CallId

and (Rtrim(WhoAcknow) is null or WhoAcknow = ' ')

or (Rtrim(WhoResolv) is null or WhoResolv = ' ') --If this condition is true, it won't matter if the others are false

Select @AckRows = @@ROWCOUNT

Select @ResolvRows = @@ROWCOUNT -- This will always return 1

If you get to update more than one CallId in the same statement, your trigger might not work properly.