No, you can't.
Because if you could, SQL wouldn't know what to put as value in the already existing records. If you didn't have any records in the table it would work without issues.
The simplest way to do this is create the column with a default and then remove the default.
ALTER TABLE dbo.MyTable ADD
MyColumn text NOT NULL CONSTRAINT DF_MyTable_MyColumn DEFAULT 'defaultValue'
ALTER TABLE dbo.MyTable
DROP CONSTRAINT DF_MyTable_MyColumn
Another alternative would be to add the column without the constraint, fill the values for all cells and add the constraint.
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